In this section, we perform our exploratory analysis to understand our dataset and identify potential issues such as missing values and outliers
First, we load in our data
eval_data <- read.csv("moneyball-evaluation-data.csv")
train_data <- read.csv("moneyball-training-data.csv")
head(eval, n=3)
##
## 1 function (expr, envir = parent.frame(), enclos = if (is.list(envir) ||
## 2 is.pairlist(envir)) parent.frame() else baseenv())
## 3 .Internal(eval(expr, envir, enclos))
head(train, n=3)
##
## 1 function (x, ...)
## 2 {
## 3 UseMethod("train")
# convert to data.frame and drop INDEX fields
eval_df <- data.frame(eval_data) |>
drop_columns("INDEX")
train_df <- data.frame(train_data) |>
drop_columns("INDEX")
The TARGET_WINS field represents the number of games a team won in a given baseball season.
This is the dependent variable (target variable) in the multiple linear regression model.
We will be trying to predict TARGET_WINS based on other team performance metrics.
We can see there are fields with null values
str(eval_df)
## 'data.frame': 259 obs. of 15 variables:
## $ TEAM_BATTING_H : int 1209 1221 1395 1539 1445 1431 1430 1385 1259 1397 ...
## $ TEAM_BATTING_2B : int 170 151 183 309 203 236 219 158 177 212 ...
## $ TEAM_BATTING_3B : int 33 29 29 29 68 53 55 42 78 42 ...
## $ TEAM_BATTING_HR : int 83 88 93 159 5 10 37 33 23 58 ...
## $ TEAM_BATTING_BB : int 447 516 509 486 95 215 568 356 466 452 ...
## $ TEAM_BATTING_SO : int 1080 929 816 914 416 377 527 609 689 584 ...
## $ TEAM_BASERUN_SB : int 62 54 59 148 NA NA 365 185 150 52 ...
## $ TEAM_BASERUN_CS : int 50 39 47 57 NA NA NA NA NA NA ...
## $ TEAM_BATTING_HBP: int NA NA NA 42 NA NA NA NA NA NA ...
## $ TEAM_PITCHING_H : int 1209 1221 1395 1539 3902 2793 1544 1626 1342 1489 ...
## $ TEAM_PITCHING_HR: int 83 88 93 159 14 20 40 39 25 62 ...
## $ TEAM_PITCHING_BB: int 447 516 509 486 257 420 613 418 497 482 ...
## $ TEAM_PITCHING_SO: int 1080 929 816 914 1123 736 569 715 734 622 ...
## $ TEAM_FIELDING_E : int 140 135 156 124 616 572 490 328 226 184 ...
## $ TEAM_FIELDING_DP: int 156 164 153 154 130 105 NA 104 132 145 ...
## - attr(*, ".internal.selfref")=<externalptr>
str(train_df)
## 'data.frame': 2276 obs. of 16 variables:
## $ TARGET_WINS : int 39 70 86 70 82 75 80 85 86 76 ...
## $ TEAM_BATTING_H : int 1445 1339 1377 1387 1297 1279 1244 1273 1391 1271 ...
## $ TEAM_BATTING_2B : int 194 219 232 209 186 200 179 171 197 213 ...
## $ TEAM_BATTING_3B : int 39 22 35 38 27 36 54 37 40 18 ...
## $ TEAM_BATTING_HR : int 13 190 137 96 102 92 122 115 114 96 ...
## $ TEAM_BATTING_BB : int 143 685 602 451 472 443 525 456 447 441 ...
## $ TEAM_BATTING_SO : int 842 1075 917 922 920 973 1062 1027 922 827 ...
## $ TEAM_BASERUN_SB : int NA 37 46 43 49 107 80 40 69 72 ...
## $ TEAM_BASERUN_CS : int NA 28 27 30 39 59 54 36 27 34 ...
## $ TEAM_BATTING_HBP: int NA NA NA NA NA NA NA NA NA NA ...
## $ TEAM_PITCHING_H : int 9364 1347 1377 1396 1297 1279 1244 1281 1391 1271 ...
## $ TEAM_PITCHING_HR: int 84 191 137 97 102 92 122 116 114 96 ...
## $ TEAM_PITCHING_BB: int 927 689 602 454 472 443 525 459 447 441 ...
## $ TEAM_PITCHING_SO: int 5456 1082 917 928 920 973 1062 1033 922 827 ...
## $ TEAM_FIELDING_E : int 1011 193 175 164 138 123 136 112 127 131 ...
## $ TEAM_FIELDING_DP: int NA 155 153 156 168 149 186 136 169 159 ...
## - attr(*, ".internal.selfref")=<externalptr>
These scatter plots give us a sense of the relationship between the each variable and TARGET_WINS. Data points are plotted where the x-axis represents the predictor variable, and the y-axis represents the number of wins. A black trend line is fitted using linear regression is showing the general direction of the relationship.
train_df %>%
gather(variable, value, -TARGET_WINS) %>%
ggplot(., aes(value, TARGET_WINS)) +
geom_point(fill = "#628B3A", color="#628B3A") +
geom_smooth(method = "lm", se = FALSE, color = "black") +
facet_wrap(~variable, scales ="free", ncol = 4) +
labs(x = element_blank(), y = "Wins")
## `geom_smooth()` using formula = 'y ~ x'
## Warning: Removed 3478 rows containing non-finite outside the scale range
## (`stat_smooth()`).
## Warning: Removed 3478 rows containing missing values or values outside the scale range
## (`geom_point()`).
Interpretation
The slope of the regression line in each facet is used to determine the strength of relationship between the independent variable represent on the x-axis vs the dependent variable y (TARGET_WINS). The steeper the slope, the stronger the relationship is between the two variables. The direction of the slope tells whether the relationship is positive or negative: - if the line is sloped to the right, it is a positive relationship meaning we can expect an increase in y as x increases - if the line is sloped to the left, it is a negative relationship meaning we can expect an decrease in y as x increases - if the trend line is flat, there is likely no meaningful relationship between that variable and TARGET_WINS.
If the points are closely clustered around the line, it suggests a stronger linear relationship. If the points are widely scattered, the variable may not strongly predict TARGET_WINS.
Positive Predictors of Wins: TEAM_BATTING_2B, TEAM_BATTING_BB, TEAM_BATTING_H, TEAM_PITCHING_BB (unexpected).
Negative Predictors of Wins: TEAM_FIELDING_E, TEAM_PITCHING_H, TEAM_PITCHING_SO.
Weak or No Influence: TEAM_BATTING_3B, TEAM_BATTING_HBP, TEAM_BATTING_SO, TEAM_FIELDING_DP, TEAM_PITCHING_HR.
plot_correlation(train_df, type = "all")
## Warning: Removed 150 rows containing missing values or values outside the scale range
## (`geom_text()`).
Interpretration
The Correlation Heatmap shows that the following variables are very highly correlated with one another - TEAM_BATTING_HR and TEAM_PITCHING_HR have a correlation value of 0.97 - TEAM_FIELDING_E and TEAM_PITCHING_H have a correlation value of 0.67 - TEAM_FIELDING_E and TEAM_BATTING_BB have a correlation value of -0.66 - TEAM_BATTING_3B and TEAM_BATTING_HR have a correlation value of -0.64 - TEAM_FIELDING_E and TEAM_BATTING_HR have a correlation value of -0.59 - TEAM_FIELDING_E and TEAM_PITCHING_HR have a correlation value of -0.57 - TEAM_BATTING_H and TEAM_BATTING_2B have a correlation value of -0.57 - TEAM_FIELDING_E and TEAM_BATTING_3B have a correlation value of 0.51 -
correlation_with_target <- cor(train_df, use = "complete.obs")["TARGET_WINS", ] %>%
sort(decreasing = TRUE) # Sort from highest to lowest correlation
print(correlation_with_target)
## TARGET_WINS TEAM_PITCHING_H TEAM_BATTING_H TEAM_BATTING_BB
## 1.00000000 0.47123431 0.46994665 0.46868793
## TEAM_PITCHING_BB TEAM_PITCHING_HR TEAM_BATTING_HR TEAM_BATTING_2B
## 0.46839882 0.42246683 0.42241683 0.31298400
## TEAM_BATTING_HBP TEAM_BASERUN_SB TEAM_BATTING_3B TEAM_BASERUN_CS
## 0.07350424 0.01483639 -0.12434586 -0.17875598
## TEAM_FIELDING_DP TEAM_BATTING_SO TEAM_PITCHING_SO TEAM_FIELDING_E
## -0.19586601 -0.22889273 -0.22936481 -0.38668800
library(ggplot2)
correlation_data <- data.frame(Variable = names(correlation_with_target), Correlation = correlation_with_target)
ggplot(correlation_data, aes(x = reorder(Variable, Correlation), y = Correlation, fill = Correlation > 0)) +
geom_bar(stat = "identity") +
coord_flip() + # Flip for better readability
labs(title = "Correlation with Target Wins", x = "Variables", y = "Correlation") +
scale_fill_manual(values = c("red", "blue")) +
theme_minimal()
### Checking Correlations and Finding Action:
Strong positive correlation (> 0.5) -> keep the variable in regression.
Strong negative correlation (< -0.5) -> Keep (inverse relationship).
Weak correlation (~0) -> Consider removing.
Two highly correlated variables (> 0.85) -> Drop one to avoid multicollinearity.
skim(eval_df)
| Name | eval_df |
| Number of rows | 259 |
| Number of columns | 15 |
| _______________________ | |
| Column type frequency: | |
| numeric | 15 |
| ________________________ | |
| Group variables | None |
Variable type: numeric
| skim_variable | n_missing | complete_rate | mean | sd | p0 | p25 | p50 | p75 | p100 | hist |
|---|---|---|---|---|---|---|---|---|---|---|
| TEAM_BATTING_H | 0 | 1.00 | 1469.39 | 150.66 | 819 | 1387.0 | 1455.0 | 1548.00 | 2170 | ▁▂▇▁▁ |
| TEAM_BATTING_2B | 0 | 1.00 | 241.32 | 49.52 | 44 | 210.0 | 239.0 | 278.50 | 376 | ▁▂▇▇▂ |
| TEAM_BATTING_3B | 0 | 1.00 | 55.91 | 27.14 | 14 | 35.0 | 52.0 | 72.00 | 155 | ▇▇▃▁▁ |
| TEAM_BATTING_HR | 0 | 1.00 | 95.63 | 56.33 | 0 | 44.5 | 101.0 | 135.50 | 242 | ▆▅▇▃▁ |
| TEAM_BATTING_BB | 0 | 1.00 | 498.96 | 120.59 | 15 | 436.5 | 509.0 | 565.50 | 792 | ▁▁▅▇▁ |
| TEAM_BATTING_SO | 18 | 0.93 | 709.34 | 243.11 | 0 | 545.0 | 686.0 | 912.00 | 1268 | ▁▃▇▇▂ |
| TEAM_BASERUN_SB | 13 | 0.95 | 123.70 | 93.39 | 0 | 59.0 | 92.0 | 151.75 | 580 | ▇▃▁▁▁ |
| TEAM_BASERUN_CS | 87 | 0.66 | 52.32 | 23.10 | 0 | 38.0 | 49.5 | 63.00 | 154 | ▂▇▃▁▁ |
| TEAM_BATTING_HBP | 240 | 0.07 | 62.37 | 12.71 | 42 | 53.5 | 62.0 | 67.50 | 96 | ▃▇▅▁▁ |
| TEAM_PITCHING_H | 0 | 1.00 | 1813.46 | 1662.91 | 1155 | 1426.5 | 1515.0 | 1681.00 | 22768 | ▇▁▁▁▁ |
| TEAM_PITCHING_HR | 0 | 1.00 | 102.15 | 57.65 | 0 | 52.0 | 104.0 | 142.50 | 336 | ▇▇▆▁▁ |
| TEAM_PITCHING_BB | 0 | 1.00 | 552.42 | 172.95 | 136 | 471.0 | 526.0 | 606.50 | 2008 | ▆▇▁▁▁ |
| TEAM_PITCHING_SO | 18 | 0.93 | 799.67 | 634.31 | 0 | 613.0 | 745.0 | 938.00 | 9963 | ▇▁▁▁▁ |
| TEAM_FIELDING_E | 0 | 1.00 | 249.75 | 230.90 | 73 | 131.0 | 163.0 | 252.00 | 1568 | ▇▁▁▁▁ |
| TEAM_FIELDING_DP | 31 | 0.88 | 146.06 | 25.88 | 69 | 131.0 | 148.0 | 164.00 | 204 | ▁▂▇▇▂ |
skim(train_df)
| Name | train_df |
| Number of rows | 2276 |
| Number of columns | 16 |
| _______________________ | |
| Column type frequency: | |
| numeric | 16 |
| ________________________ | |
| Group variables | None |
Variable type: numeric
| skim_variable | n_missing | complete_rate | mean | sd | p0 | p25 | p50 | p75 | p100 | hist |
|---|---|---|---|---|---|---|---|---|---|---|
| TARGET_WINS | 0 | 1.00 | 80.79 | 15.75 | 0 | 71.0 | 82.0 | 92.00 | 146 | ▁▁▇▅▁ |
| TEAM_BATTING_H | 0 | 1.00 | 1469.27 | 144.59 | 891 | 1383.0 | 1454.0 | 1537.25 | 2554 | ▁▇▂▁▁ |
| TEAM_BATTING_2B | 0 | 1.00 | 241.25 | 46.80 | 69 | 208.0 | 238.0 | 273.00 | 458 | ▁▆▇▂▁ |
| TEAM_BATTING_3B | 0 | 1.00 | 55.25 | 27.94 | 0 | 34.0 | 47.0 | 72.00 | 223 | ▇▇▂▁▁ |
| TEAM_BATTING_HR | 0 | 1.00 | 99.61 | 60.55 | 0 | 42.0 | 102.0 | 147.00 | 264 | ▇▆▇▅▁ |
| TEAM_BATTING_BB | 0 | 1.00 | 501.56 | 122.67 | 0 | 451.0 | 512.0 | 580.00 | 878 | ▁▁▇▇▁ |
| TEAM_BATTING_SO | 102 | 0.96 | 735.61 | 248.53 | 0 | 548.0 | 750.0 | 930.00 | 1399 | ▁▆▇▇▁ |
| TEAM_BASERUN_SB | 131 | 0.94 | 124.76 | 87.79 | 0 | 66.0 | 101.0 | 156.00 | 697 | ▇▃▁▁▁ |
| TEAM_BASERUN_CS | 772 | 0.66 | 52.80 | 22.96 | 0 | 38.0 | 49.0 | 62.00 | 201 | ▃▇▁▁▁ |
| TEAM_BATTING_HBP | 2085 | 0.08 | 59.36 | 12.97 | 29 | 50.5 | 58.0 | 67.00 | 95 | ▂▇▇▅▁ |
| TEAM_PITCHING_H | 0 | 1.00 | 1779.21 | 1406.84 | 1137 | 1419.0 | 1518.0 | 1682.50 | 30132 | ▇▁▁▁▁ |
| TEAM_PITCHING_HR | 0 | 1.00 | 105.70 | 61.30 | 0 | 50.0 | 107.0 | 150.00 | 343 | ▇▇▆▁▁ |
| TEAM_PITCHING_BB | 0 | 1.00 | 553.01 | 166.36 | 0 | 476.0 | 536.5 | 611.00 | 3645 | ▇▁▁▁▁ |
| TEAM_PITCHING_SO | 102 | 0.96 | 817.73 | 553.09 | 0 | 615.0 | 813.5 | 968.00 | 19278 | ▇▁▁▁▁ |
| TEAM_FIELDING_E | 0 | 1.00 | 246.48 | 227.77 | 65 | 127.0 | 159.0 | 249.25 | 1898 | ▇▁▁▁▁ |
| TEAM_FIELDING_DP | 286 | 0.87 | 146.39 | 26.23 | 52 | 131.0 | 149.0 | 164.00 | 228 | ▁▂▇▆▁ |
summary(train_df)
## TARGET_WINS TEAM_BATTING_H TEAM_BATTING_2B TEAM_BATTING_3B
## Min. : 0.00 Min. : 891 Min. : 69.0 Min. : 0.00
## 1st Qu.: 71.00 1st Qu.:1383 1st Qu.:208.0 1st Qu.: 34.00
## Median : 82.00 Median :1454 Median :238.0 Median : 47.00
## Mean : 80.79 Mean :1469 Mean :241.2 Mean : 55.25
## 3rd Qu.: 92.00 3rd Qu.:1537 3rd Qu.:273.0 3rd Qu.: 72.00
## Max. :146.00 Max. :2554 Max. :458.0 Max. :223.00
##
## TEAM_BATTING_HR TEAM_BATTING_BB TEAM_BATTING_SO TEAM_BASERUN_SB
## Min. : 0.00 Min. : 0.0 Min. : 0.0 Min. : 0.0
## 1st Qu.: 42.00 1st Qu.:451.0 1st Qu.: 548.0 1st Qu.: 66.0
## Median :102.00 Median :512.0 Median : 750.0 Median :101.0
## Mean : 99.61 Mean :501.6 Mean : 735.6 Mean :124.8
## 3rd Qu.:147.00 3rd Qu.:580.0 3rd Qu.: 930.0 3rd Qu.:156.0
## Max. :264.00 Max. :878.0 Max. :1399.0 Max. :697.0
## NA's :102 NA's :131
## TEAM_BASERUN_CS TEAM_BATTING_HBP TEAM_PITCHING_H TEAM_PITCHING_HR
## Min. : 0.0 Min. :29.00 Min. : 1137 Min. : 0.0
## 1st Qu.: 38.0 1st Qu.:50.50 1st Qu.: 1419 1st Qu.: 50.0
## Median : 49.0 Median :58.00 Median : 1518 Median :107.0
## Mean : 52.8 Mean :59.36 Mean : 1779 Mean :105.7
## 3rd Qu.: 62.0 3rd Qu.:67.00 3rd Qu.: 1682 3rd Qu.:150.0
## Max. :201.0 Max. :95.00 Max. :30132 Max. :343.0
## NA's :772 NA's :2085
## TEAM_PITCHING_BB TEAM_PITCHING_SO TEAM_FIELDING_E TEAM_FIELDING_DP
## Min. : 0.0 Min. : 0.0 Min. : 65.0 Min. : 52.0
## 1st Qu.: 476.0 1st Qu.: 615.0 1st Qu.: 127.0 1st Qu.:131.0
## Median : 536.5 Median : 813.5 Median : 159.0 Median :149.0
## Mean : 553.0 Mean : 817.7 Mean : 246.5 Mean :146.4
## 3rd Qu.: 611.0 3rd Qu.: 968.0 3rd Qu.: 249.2 3rd Qu.:164.0
## Max. :3645.0 Max. :19278.0 Max. :1898.0 Max. :228.0
## NA's :102 NA's :286
summary(eval_df)
## TEAM_BATTING_H TEAM_BATTING_2B TEAM_BATTING_3B TEAM_BATTING_HR
## Min. : 819 Min. : 44.0 Min. : 14.00 Min. : 0.00
## 1st Qu.:1387 1st Qu.:210.0 1st Qu.: 35.00 1st Qu.: 44.50
## Median :1455 Median :239.0 Median : 52.00 Median :101.00
## Mean :1469 Mean :241.3 Mean : 55.91 Mean : 95.63
## 3rd Qu.:1548 3rd Qu.:278.5 3rd Qu.: 72.00 3rd Qu.:135.50
## Max. :2170 Max. :376.0 Max. :155.00 Max. :242.00
##
## TEAM_BATTING_BB TEAM_BATTING_SO TEAM_BASERUN_SB TEAM_BASERUN_CS
## Min. : 15.0 Min. : 0.0 Min. : 0.0 Min. : 0.00
## 1st Qu.:436.5 1st Qu.: 545.0 1st Qu.: 59.0 1st Qu.: 38.00
## Median :509.0 Median : 686.0 Median : 92.0 Median : 49.50
## Mean :499.0 Mean : 709.3 Mean :123.7 Mean : 52.32
## 3rd Qu.:565.5 3rd Qu.: 912.0 3rd Qu.:151.8 3rd Qu.: 63.00
## Max. :792.0 Max. :1268.0 Max. :580.0 Max. :154.00
## NA's :18 NA's :13 NA's :87
## TEAM_BATTING_HBP TEAM_PITCHING_H TEAM_PITCHING_HR TEAM_PITCHING_BB
## Min. :42.00 Min. : 1155 Min. : 0.0 Min. : 136.0
## 1st Qu.:53.50 1st Qu.: 1426 1st Qu.: 52.0 1st Qu.: 471.0
## Median :62.00 Median : 1515 Median :104.0 Median : 526.0
## Mean :62.37 Mean : 1813 Mean :102.1 Mean : 552.4
## 3rd Qu.:67.50 3rd Qu.: 1681 3rd Qu.:142.5 3rd Qu.: 606.5
## Max. :96.00 Max. :22768 Max. :336.0 Max. :2008.0
## NA's :240
## TEAM_PITCHING_SO TEAM_FIELDING_E TEAM_FIELDING_DP
## Min. : 0.0 Min. : 73.0 Min. : 69.0
## 1st Qu.: 613.0 1st Qu.: 131.0 1st Qu.:131.0
## Median : 745.0 Median : 163.0 Median :148.0
## Mean : 799.7 Mean : 249.7 Mean :146.1
## 3rd Qu.: 938.0 3rd Qu.: 252.0 3rd Qu.:164.0
## Max. :9963.0 Max. :1568.0 Max. :204.0
## NA's :18 NA's :31
train_df %>%
summarise(across(where(is.numeric), list(mean = mean, median = median, sd = sd), na.rm = TRUE))
## Warning: There was 1 warning in `summarise()`.
## ℹ In argument: `across(...)`.
## Caused by warning:
## ! The `...` argument of `across()` is deprecated as of dplyr 1.1.0.
## Supply arguments directly to `.fns` through an anonymous function instead.
##
## # Previously
## across(a:b, mean, na.rm = TRUE)
##
## # Now
## across(a:b, \(x) mean(x, na.rm = TRUE))
The summarise(across()) function above is calculating the mean, median, ans standard deviation for all the numerical variables.
Before fitting a multiple linear regression (MLR) model, we analyze the dataset for potential issues such as missing values, extreme outliers, multicollinearity, and variable distributions.
Here’s what we can infer from the summary statistics:
Range: 0 to 146 wins
Mean: ~80.79 wins
Median: 82 wins
Distribution: The min value of 0 and max of 146 suggest some potential outliers or erroneous data points, since most teams win between 50-110 games in a season.
boxplot(train_df$TARGET_WINS, main="Distribution of Team Wins", ylab="Wins", col="lightblue")
Actionable Steps:
Some variables have a significant number of missing values. In particular:
Additionally, four variables have values of zero (0) reported that appear suspicious. In particular: - TEAM_BATTING_SO & TEAM_PITHCING_SO have the same rows entered as zero suggesting that data may not have been available for these entries. - TEAM_BATTING_HR & TEAM_PITHCING_HR have the same rows entered as zero suggesting that data may not have been available for these entries.
Actionable Steps:
Impute missing values (using mean/median or regression techniques).
Consider removing TEAM_BATTING_HBP and TEAM_BASERUN_CS if they are highly incomplete and do not contribute much.
Several variables have extreme max values that seem unrealistic. Specifically:
TEAM_PITCHING_H (Max = 30,132) <- Likely an error since typical values range from 1,200 - 1,700.
TEAM_PITCHING_SO (Max = 19,278) <- Suspiciously high (typical range: 500 - 1,500).
TEAM_PITCHING_BB (Max = 3,645) <- Very high (typical range: 300 - 700).
TEAM_FIELDING_E (Max = 1,898) <- Likely an error since the normal range is ~ 70-200.
Actionable Steps:
Check for data entry errors.
Remove extreme outliers if they distort model performance.
Batting Variables: TEAM_BATTING_H, TEAM_BATTING_2B, TEAM_BATTING_3B, TEAM_BATTING_HR, TEAM_BATTING_BB are likely strong predictors of team wins. Note that TEAM_BATTING_H includes TEAM_BATTING_2B, TEAM_BATTING_3B, TEAM_BATTING_HR data.
Pitching Variables: TEAM_PITCHING_H, TEAM_PITCHING_HR, TEAM_PITCHING_BB, TEAM_PITCHING_SO will impact defensive strength.
Fielding Variables: TEAM_FIELDING_E (errors) and TEAM_FIELDING_DP (double plays) may have a weaker impact compared to batting and pitching.
Actionable Steps:
TEAM_PITCHING_H, TEAM_PITCHING_HR, and TEAM_PITCHING_BB may be highly correlated, which can cause multicollinearity in the regression model.
TEAM_BATTING_H, TEAM_BATTING_2B, and TEAM_BATTING_3B may also be strongly correlated since total hits include doubles and triples.
Handle Missing Values
Consider dropping or imputing variables with too many missing values (e.g., TEAM_BATTING_HBP).
Impute TEAM_BASERUN_SB, TEAM_BASERUN_CS, and TEAM_FIELDING_DP appropriately.
Remove highly unrealistic values in pitching, fielding, and errors.
Use Variance Inflation Factor (VIF) to detect multicollinearity and drop redundant features.
Consider derived metrics like batting average (H/AB), on-base percentage (OBP), or earned run average (ERA) instead of raw counts.
Conclusion
The dataset contains inconsistencies, missing values, and extreme outliers that need to be addressed before fitting an MLR model.
Once cleaned, feature selection and multicollinearity checks will be essential to ensure a robust and interpretable model for predicting team wins.
This box plot provides valuable insights into the distribution of team wins in the training dataset.
Here’s what we can infer:
The thick horizontal line inside the box represents the median (~82 wins).
The box itself (Interquartile Range - IQR) shows the middle 50% of the data, which seems to range roughly from 70 to 92 wins.
Low-end outliers (~0-40 wins): There are several small circles (outliers) below the lower whisker.
High-end outliers (~110-146 wins): There are some outliers above the upper whisker, but visually fewer than the low-end.
These low-win teams might be problematic for modeling because they could represent incomplete or missing data.
Potential data entry issues (e.g., a team with 0 wins) should be checked.
If extreme values skew the regression, we might need transformations (log scaling)
The box is fairly centered, suggesting a roughly symmetric distribution.
hist(train_df$TARGET_WINS,
main = "Histogram of Team Wins with Density Curve",
xlab = "Wins",
col = "lightgray",
border = "black",
breaks = 20,
probability = TRUE) # Converts y-axis to density
lines(density(train_df$TARGET_WINS, na.rm = TRUE),
col = "red",
lwd = 2) # Adds a red density curve
missing_values <- train_df %>%
summarise(across(everything(), ~ sum(is.na(.)))) %>%
pivot_longer(cols = everything(), names_to = "Variable", values_to = "Missing_Count")
print(missing_values)
## # A tibble: 16 × 2
## Variable Missing_Count
## <chr> <int>
## 1 TARGET_WINS 0
## 2 TEAM_BATTING_H 0
## 3 TEAM_BATTING_2B 0
## 4 TEAM_BATTING_3B 0
## 5 TEAM_BATTING_HR 0
## 6 TEAM_BATTING_BB 0
## 7 TEAM_BATTING_SO 102
## 8 TEAM_BASERUN_SB 131
## 9 TEAM_BASERUN_CS 772
## 10 TEAM_BATTING_HBP 2085
## 11 TEAM_PITCHING_H 0
## 12 TEAM_PITCHING_HR 0
## 13 TEAM_PITCHING_BB 0
## 14 TEAM_PITCHING_SO 102
## 15 TEAM_FIELDING_E 0
## 16 TEAM_FIELDING_DP 286
ggplot(missing_values, aes(y = reorder(Variable, Missing_Count), x = Missing_Count, fill = Missing_Count > 0)) +
geom_col() +
labs(title = "Missing Values in Training Dataset",
x = "Number of Missing Values",
y = "Variables") +
scale_fill_manual(values = c("gray", "red"), labels = c("No Missing", "Has Missing")) +
theme_minimal()
### Strategy for Missing Values Column wise:
There are four main options for handling missing values:
If the TARGET_WINS column has missing values, remove those rows since we can’t predict missing outcomes.
train_df <- train_df %>% filter(!is.na(TARGET_WINS))
If a column has too many missing values (e.g., >50% missing), it may be better to remove it.
The TEAM_BATTING_HBP column is mostly empty and not critical.
train_df <- train_df[, !names(train_df) %in% "TEAM_BATTING_HBP"]
The variables TEAM_BATTING_SO, TEAM_PITCHING_SO, TEAM_BATTING_HR, and TEAM_PITCHING_HR included several observations with a value of zero. As the rows were the same for both variables, it appears that these may also be missing observations as the likelihood that a team’s batters did not have a single strikeout nor did their pitchers pitch a single strikeout over the course of a 162 game season is highly unlikely. We will therefore treat these as missing observations and impute using the median value which is more robust to outliers then using the mean values. We will also drop the single row where the team did not win a single game, as this is also suspicious.
# Convert dubious stats to NAs for pitching
# and drop unnecessary columns
train_df <- train_df |>
mutate(
TEAM_BATTING_SO = if_else(TEAM_BATTING_SO > 0, TEAM_BATTING_SO, NA_integer_),
TEAM_PITCHING_SO = if_else(TEAM_PITCHING_SO > 0, TEAM_PITCHING_SO, NA_integer_),
TEAM_BATTING_HR = if_else(TEAM_BATTING_HR > 0, TEAM_BATTING_HR, NA_integer_),
TEAM_PITCHING_HR = if_else(TEAM_PITCHING_HR > 0, TEAM_PITCHING_HR, NA_integer_)
) |>
filter(TARGET_WINS > 0)
Next, we will address the remaining missing values. We will weight several options
Missing Values Cause Errors in Regression Models
lm() in R cannot handle missing values and will return an error if NAs exist in predictor variables.
Removing missing values ensures that the model runs smoothly without interruptions.
Pros
Cons
If missing values are not randomly distributed, removing them may bias the dataset.
Reduced sample size
Conclusion
Instead of na.omit(), imputation methods (like mean/median filling) may be better for handling missing data.
Removing rows with missing data is sometimes not the best approach, especially if a large portion of data is lost. We will therefore explore other methods.
Instead of removing missing values, fill them with the mean:
train_df_mean <- train_df
mean_val <- colMeans(train_df, na.rm = TRUE)
# Impute using Means
for(i in colnames(train_df))
train_df[,i][is.na(train_df[,i])] <- mean_val[i]
plot_qq(train_df, sampled_rows = 1000L)
# Missing values per column
colSums(is.na(train_df_mean))
## TARGET_WINS TEAM_BATTING_H TEAM_BATTING_2B TEAM_BATTING_3B
## 0 0 0 0
## TEAM_BATTING_HR TEAM_BATTING_BB TEAM_BATTING_SO TEAM_BASERUN_SB
## 14 0 121 131
## TEAM_BASERUN_CS TEAM_PITCHING_H TEAM_PITCHING_HR TEAM_PITCHING_BB
## 772 0 14 0
## TEAM_PITCHING_SO TEAM_FIELDING_E TEAM_FIELDING_DP
## 121 0 285
Pros
Cons Not usually recommended for serious analyses because it reduces the variance and doesn’t account for the uncertainty in imputations.
Instead of removing missing values, fill them with the median:
library(miscTools)
train_df_median <- train_df
median_val <- colMedians(train_df, na.rm = TRUE)
# Impute using medians
for(i in colnames(train_df))
train_df_median[,i][is.na(train_df_median[,i])] <- as.integer(median_val[i])
plot_qq(train_df_median, sampled_rows = 1000L)
# Missing values per column
colSums(is.na(train_df_median))
## TARGET_WINS TEAM_BATTING_H TEAM_BATTING_2B TEAM_BATTING_3B
## 0 0 0 0
## TEAM_BATTING_HR TEAM_BATTING_BB TEAM_BATTING_SO TEAM_BASERUN_SB
## 0 0 0 0
## TEAM_BASERUN_CS TEAM_PITCHING_H TEAM_PITCHING_HR TEAM_PITCHING_BB
## 0 0 0 0
## TEAM_PITCHING_SO TEAM_FIELDING_E TEAM_FIELDING_DP
## 0 0 0
Pros
Cons
Multivariate Imputation by Chained Equations (MICE) is a type of Multiple Imputation.
train_df_mice <- mice(data=train_df, m=30, maxit=10, seed=12345, print=FALSE)
train_df_clean <- train_df_mice$data
sum(is.na(train_df_clean)) # Total missing values
## [1] 0
Pros
Cons
We split the original training dataset into a training and testing dataset in order to test the strength of our model without testing the model on data that the model has already seen. The training dataset will hold 75% of our original observations, while the test dataset will hold 25%.
smp_size <- floor(0.75 * nrow(train_df_clean))
nrow(train_df_clean)
## [1] 2275
## set the seed to make your partition reproducible
set.seed(123)
train_ind <- sample(seq_len(nrow(train_df_clean)), size = smp_size)
stp75_train_df <- train_df_clean[train_ind, ]
stp25_test_df <- train_df_clean[-train_ind, ]
The Residuals vs. Fitted and QQ Plots show a fairly linear pattern, while Scale-Location plot suggest Homoscedasticity. However, the Residuals vs Leverage plot reveals the presence of some outliers.
stp_model_full <- lm(TARGET_WINS ~ ., data = stp75_train_df)
summary(stp_model_full)
##
## Call:
## lm(formula = TARGET_WINS ~ ., data = stp75_train_df)
##
## Residuals:
## Min 1Q Median 3Q Max
## -49.316 -8.649 0.045 8.329 57.170
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 17.6962800 6.3050518 2.807 0.005063 **
## TEAM_BATTING_H 0.0514744 0.0044700 11.516 < 2e-16 ***
## TEAM_BATTING_2B -0.0224439 0.0106248 -2.112 0.034798 *
## TEAM_BATTING_3B 0.0672376 0.0199897 3.364 0.000786 ***
## TEAM_BATTING_HR 0.0285624 0.0317578 0.899 0.368578
## TEAM_BATTING_BB 0.0066057 0.0067404 0.980 0.327224
## TEAM_BATTING_SO -0.0034806 0.0029004 -1.200 0.230288
## TEAM_BASERUN_SB 0.0278236 0.0051813 5.370 8.96e-08 ***
## TEAM_BASERUN_CS -0.0136696 0.0181769 -0.752 0.452135
## TEAM_PITCHING_H -0.0008407 0.0004833 -1.740 0.082111 .
## TEAM_PITCHING_HR 0.0239234 0.0289737 0.826 0.409095
## TEAM_PITCHING_BB 0.0018163 0.0047792 0.380 0.703961
## TEAM_PITCHING_SO 0.0021152 0.0009863 2.144 0.032138 *
## TEAM_FIELDING_E -0.0223272 0.0028410 -7.859 6.84e-15 ***
## TEAM_FIELDING_DP -0.1065083 0.0149272 -7.135 1.43e-12 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 13.07 on 1691 degrees of freedom
## Multiple R-squared: 0.3031, Adjusted R-squared: 0.2973
## F-statistic: 52.53 on 14 and 1691 DF, p-value: < 2.2e-16
# check for outliers using cooks-distance plot
plot(stp_model_full, which = 4, id.n = 8)
# get points of influence
influence <- influence.measures(stp_model_full)
influential_points <- influence$infmat
cooks_d <- influence$infmat[, "cook.d"]
max_influence_index <- which.max(cooks_d)
The observation with index 2135 is particularly problematic. A closer examination reveals TEAM_PITCHING_SO is almost 75% higher as the next highest value (19278 vs 12758). TEAM_PITCHING_H is also unusually high for this year.
influential_data_point <- stp75_train_df[max_influence_index, ]
print(influential_data_point)
## TARGET_WINS TEAM_BATTING_H TEAM_BATTING_2B TEAM_BATTING_3B TEAM_BATTING_HR
## 2135 41 992 263 20 100.2729
## TEAM_BATTING_BB TEAM_BATTING_SO TEAM_BASERUN_SB TEAM_BASERUN_CS
## 2135 142 952 124.82 52.83899
## TEAM_PITCHING_H TEAM_PITCHING_HR TEAM_PITCHING_BB TEAM_PITCHING_SO
## 2135 20088 106.3998 2876 19278
## TEAM_FIELDING_E TEAM_FIELDING_DP
## 2135 952 146.3879
We will drop this row since it has such a high leverage on the model.
# remove outlier
stp75_train_df <- stp75_train_df |>
filter(TEAM_PITCHING_SO < 15000)
# confirm outliers
stp_model_full <- lm(TARGET_WINS ~ ., data = stp75_train_df)
plot(stp_model_full, which = 4, id.n = 8)
Charting the plot shows another point (202) with high influence. This record has a TEAM_PITCHING_SO that is more than twice the next closest value.
# get points of influence
influence <- influence.measures(stp_model_full)
influential_points <- influence$infmat
cooks_d <- influence$infmat[, "cook.d"]
max_influence_index <- which.max(cooks_d)
influential_data_point <- stp75_train_df[max_influence_index, ]
print(influential_data_point)
## TARGET_WINS TEAM_BATTING_H TEAM_BATTING_2B TEAM_BATTING_3B TEAM_BATTING_HR
## 202 108 1188 338 0 100.2729
## TEAM_BATTING_BB TEAM_BATTING_SO TEAM_BASERUN_SB TEAM_BASERUN_CS
## 202 270 945 124.82 52.83899
## TEAM_PITCHING_H TEAM_PITCHING_HR TEAM_PITCHING_BB TEAM_PITCHING_SO
## 202 16038 106.3998 3645 12758
## TEAM_FIELDING_E TEAM_FIELDING_DP
## 202 716 146.3879
# remove outlier at row 202
stp75_train_df <- stp75_train_df[-c(202), ]
# confirm outliers
stp_model_full <- lm(TARGET_WINS ~ ., data = stp75_train_df)
plot(stp_model_full, which = 4, id.n = 8)
As the Cooks Distance (Di) value is less than 0.5 for all of the remaining outliers appear, they are not significantly influential and can be left in our dataset.
TEAM_BATTING_H (Hits): More hits increase the chances of scoring.
TEAM_BATTING_HR (Home Runs): Home runs are a major contributor to runs.
TEAM_PITCHING_SO (Strikeouts): More strikeouts reduce opponent scoring.
TEAM_FIELDING_E (Errors): More errors lead to more opponent runs (negative predictor).
Why exclude some variables?
TEAM_BASERUN_SB (Stolen Bases): Limited impact on overall wins.
TEAM_PITCHING_BB (Walks Allowed): May not be as predictive when combined with strikeouts.
base_model <- lm(TARGET_WINS ~ TEAM_BATTING_H + TEAM_BATTING_HR + TEAM_PITCHING_SO + TEAM_FIELDING_E, data = stp75_train_df)
# View model summary
summary(base_model)
##
## Call:
## lm(formula = TARGET_WINS ~ TEAM_BATTING_H + TEAM_BATTING_HR +
## TEAM_PITCHING_SO + TEAM_FIELDING_E, data = stp75_train_df)
##
## Residuals:
## Min 1Q Median 3Q Max
## -49.067 -9.426 0.048 9.562 50.510
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 12.5084602 4.3458592 2.878 0.00405 **
## TEAM_BATTING_H 0.0501497 0.0028185 17.793 < 2e-16 ***
## TEAM_BATTING_HR -0.0007802 0.0077359 -0.101 0.91968
## TEAM_PITCHING_SO -0.0001984 0.0014372 -0.138 0.89021
## TEAM_FIELDING_E -0.0215516 0.0020224 -10.656 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 13.67 on 1699 degrees of freedom
## Multiple R-squared: 0.2297, Adjusted R-squared: 0.2279
## F-statistic: 126.7 on 4 and 1699 DF, p-value: < 2.2e-16
1- Intercept (8.05)
When all predictor variables (TEAM_BATTING_H, TEAM_BATTING_HR, TEAM_PITCHING_SO, TEAM_FIELDING_E) are zero, a team is expected to have 8.05 wins. Significant (p = 0.04) → The intercept is meaningful in this context.
2- TEAM_BATTING_H (Hits) → 0.05 (p < 2e-16 *)
positive & highly significant Interpretation: For every additional hit, the team is expected to win 0.052 more games. Example: If a team gets 100 more hits in a season, they would be expected to win ~5 more games (100 × 0.05). Conclusion: More hits lead to more wins, which is expected in baseball.
3- TEAM_BATTING_HR (Home Runs) → -0.008 (p = 0.26026 - Not Significant) Negative (unexpected) & not significant
Interpretation: More home runs slightly decrease wins, but the effect is very small and not statistically significant. Example: Hitting 100 more home runs would decrease wins by 008, which doesn’t make sense. We should consider removing this variable.
Possible Issues: Multicollinearity: Home runs may be highly correlated with other batting stats (like hits or doubles), causing misleading coefficients. Outliers/Bad Teams: Some losing teams hit a lot of home runs but still lost, skewing results.
Solution: Check VIF (Variance Inflation Factor) for multicollinearity. Add an interaction term (e.g., TEAM_BATTING_HR * TEAM_BATTING_BB). Consider removing this variable if it remains insignificant.
4_ TEAM_PITCHING_SO (Strikeouts by Pitchers) → +0.0025 (p = 0.005) Weakly positive, borderline significant
Interpretation: More strikeouts slightly increase wins, but the effect is very small tough statistically significant (p = 0.005). Example: If a team strikes out 500 more batters in a season, they would win 0.5 more games.
Conclusion: Strikeouts help teams win, but they are not the strongest predictor of wins. The effect might be hidden by other defensive factors (e.g., walks, home runs allowed).
Solution: Consider adding walks (TEAM_PITCHING_BB) or earned run average (ERA) to capture pitching effectiveness better.
5- TEAM_FIELDING_E (Errors) → -0.023 (p < 2e-16 *) Negative & highly significant
Interpretation: For every additional error, a team is expected to lose 0.023 more games. Example: A team with 50 more errors in a season would lose ~1 more game (50 × 0.021). Conclusion: More errors directly hurt a team’s chances of winning, which makes sense in baseball.
Residual Std. Error 13.78 The average prediction error is ~13.78 wins.
Adjusted R² 0.2349 The model explains 23.5% of variance in TARGET_WINS (not very strong).
F-Statistic 175.6 (p < 2.2e-16) The overall model is statistically significant.
p < 2.2e-16 → The probability of getting this result by random chance is essentially 0 (very small).
At least one of the predictor variables in the model significantly affects TARGET_WINS.
If the p-value is very small (< 0.05), we reject the null hypothesis that “none of the independent variables explain wins.”
Our model as a whole is meaningful and explains a significant amount of variation in team wins.
At least one of our predictors (TEAM_BATTING_H, TEAM_BATTING_HR, TEAM_PITCHING_SO, TEAM_FIELDING_E) is statistically significant in predicting wins.
The F-statistic of 175.6 (p < 2.2e-16) means our model is highly statistically significant. This confirms that at least one of our variables—such as home runs, hits, strikeouts, or errors—has a real impact on predicting wins.
However, we still need to check which specific variables are the most meaningful (p-values of individual coefficients) and whether we can improve the model further.
Multicollinearity occurs when predictor variables are highly correlated, leading to unstable coefficients and inflated standard errors.
We use the Variance Inflation Factor (VIF) test. A VIF > 5 suggests multicollinearity, and VIF > 10 is a strong sign of redundancy.
library(car)
vif(base_model)
## TEAM_BATTING_H TEAM_BATTING_HR TEAM_PITCHING_SO TEAM_FIELDING_E
## 1.383400 2.013796 1.541450 1.824341
Since all VIF values are below 5, there is no significant multicollinearity in the model. This means:
Each predictor contributes unique information to explaining TARGET_WINS. Regression coefficients are stable, and we do not need to remove any variables due to multicollinearity. The model is not distorted by highly correlated predictors.
If a team hits more home runs and draws more walks, they likely score more runs. We test if walks amplify the impact of home runs on wins.
interaction_model <- lm(TARGET_WINS ~ TEAM_BATTING_H + TEAM_BATTING_HR + TEAM_BATTING_BB +
TEAM_BATTING_HR:TEAM_BATTING_BB +
TEAM_PITCHING_SO + TEAM_FIELDING_E,
data = stp75_train_df)
summary(interaction_model)
##
## Call:
## lm(formula = TARGET_WINS ~ TEAM_BATTING_H + TEAM_BATTING_HR +
## TEAM_BATTING_BB + TEAM_BATTING_HR:TEAM_BATTING_BB + TEAM_PITCHING_SO +
## TEAM_FIELDING_E, data = stp75_train_df)
##
## Residuals:
## Min 1Q Median 3Q Max
## -54.107 -8.711 0.072 9.142 52.211
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 27.7763894 5.3669583 5.175 2.54e-07 ***
## TEAM_BATTING_H 0.0479284 0.0027582 17.377 < 2e-16 ***
## TEAM_BATTING_HR -0.2409574 0.0282343 -8.534 < 2e-16 ***
## TEAM_BATTING_BB -0.0205540 0.0054979 -3.739 0.000191 ***
## TEAM_PITCHING_SO -0.0008999 0.0014301 -0.629 0.529258
## TEAM_FIELDING_E -0.0235317 0.0023355 -10.076 < 2e-16 ***
## TEAM_BATTING_HR:TEAM_BATTING_BB 0.0004341 0.0000507 8.562 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 13.33 on 1697 degrees of freedom
## Multiple R-squared: 0.2685, Adjusted R-squared: 0.2659
## F-statistic: 103.8 on 6 and 1697 DF, p-value: < 2.2e-16
Model now includes an interaction term (TEAM_BATTING_HR:TEAM_BATTING_BB) to see if walks (BB) affect the impact of home runs (HR) on wins. Let’s break down the results.
The average prediction error is ±13.31 wins, slightly better than before (13.78). The model explains 27.2% of the variance in wins (up from 23.6% in the original model). Similar to R², meaning additional predictors added value to the model. The model as a whole is statistically significant (at least one predictor explains wins).
More Home Runs (HR) Alone → Fewer Wins (Unexpected): The negative coefficient (-0.1650) on TEAM_BATTING_HR suggests that hitting more home runs alone does not necessarily lead to more wins.
Walks (BB) Alone Have a Weak Impact on Wins: The coefficient for TEAM_BATTING_BB is negative (-0.0074) and not statistically significant (p = 0.1387). This means that walks alone do not have a strong impact on wins.
The Interaction Term (TEAM_BATTING_HR * TEAM_BATTING_BB) is Highly Significant (p = 3.39e-10) Positive Coefficient (+0.000301)
Teams that hit home runs AND get on base with walks tend to win more games. This confirms that home runs are more valuable when combined with walks.
We want to see how home runs (TEAM_BATTING_HR) and walks (TEAM_BATTING_BB) impact wins (TARGET_WINS) together.
library(ggplot2)
ggplot(stp75_train_df, aes(x = TEAM_BATTING_HR, y = TARGET_WINS, color = TEAM_BATTING_BB)) +
geom_point(alpha = 0.7) +
geom_smooth(method = "lm", se = FALSE, color = "black") +
scale_color_gradient(low = "blue", high = "red") +
labs(title = "Interaction Effect of Home Runs and Walks on Wins",
x = "Home Runs",
y = "Wins",
color = "Walks (BB)") +
theme_minimal()
## `geom_smooth()` using formula = 'y ~ x'
Red (high walks) teams should have higher wins for the same HRs. Blue
(low walks) teams may not benefit as much from HRs. The trendline is
steeper for teams with more walks, confirming that walks amplify HR
impact.
Doubles (2B) are a strong indicator of offensive power and often correlate with scoring more runs. If a team doesn’t hit home runs, but hits many doubles, it can still score efficiently.
improved_model <- lm(TARGET_WINS ~ TEAM_BATTING_H + TEAM_BATTING_HR + TEAM_BATTING_BB +
TEAM_BATTING_2B + TEAM_BATTING_HR:TEAM_BATTING_BB +
TEAM_PITCHING_SO + TEAM_FIELDING_E,
data = stp75_train_df)
summary(improved_model)
##
## Call:
## lm(formula = TARGET_WINS ~ TEAM_BATTING_H + TEAM_BATTING_HR +
## TEAM_BATTING_BB + TEAM_BATTING_2B + TEAM_BATTING_HR:TEAM_BATTING_BB +
## TEAM_PITCHING_SO + TEAM_FIELDING_E, data = stp75_train_df)
##
## Residuals:
## Min 1Q Median 3Q Max
## -56.567 -8.618 0.106 8.699 55.401
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 2.242e+01 5.560e+00 4.033 5.76e-05 ***
## TEAM_BATTING_H 5.686e-02 3.736e-03 15.221 < 2e-16 ***
## TEAM_BATTING_HR -2.337e-01 2.821e-02 -8.284 2.38e-16 ***
## TEAM_BATTING_BB -1.997e-02 5.482e-03 -3.643 0.000277 ***
## TEAM_BATTING_2B -3.717e-02 1.053e-02 -3.531 0.000425 ***
## TEAM_PITCHING_SO 3.299e-04 1.467e-03 0.225 0.822121
## TEAM_FIELDING_E -2.588e-02 2.421e-03 -10.691 < 2e-16 ***
## TEAM_BATTING_HR:TEAM_BATTING_BB 4.299e-04 5.054e-05 8.505 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 13.28 on 1696 degrees of freedom
## Multiple R-squared: 0.2738, Adjusted R-squared: 0.2708
## F-statistic: 91.37 on 7 and 1696 DF, p-value: < 2.2e-16
The average prediction error is ±13.42 wins (slightly better than before). The model explains 27.9% of the variance in wins (slightly better than 27.2% in the previous model). Adjusted for number of predictors (still an improvement from before). The model is statistically significant overall.
Model performance improved slightly (R² increased from 27.2% → 27.9%). Doubles (TEAM_BATTING_2B) unexpectedly have a negative impact on wins.
Possible issue: Doubles may be highly correlated with other batting stats (e.g., hits, HRs). Solution: We will Check multicollinearity (VIF) or add an interaction term (e.g., TEAM_BATTING_2B * TEAM_BATTING_H). HRs alone are still negative (-0.1612), but the interaction term remains strong. Conclusion: HRs are only useful when paired with walks.
Adding TEAM_BATTING_2B slightly improves model performance
R² increased from 27.2% → 27.9% (small improvement). Residual Standard Error decreased from 13.31 → 13.25 (better fit). Unexpected negative coefficient for doubles (-0.0429, p = 3.09e-06)
Suggests that more doubles lead to fewer wins, which is counterintuitive. Possible reasons: Multicollinearity with TEAM_BATTING_H (hits). Bad teams might hit many doubles but still lose. Interaction term (TEAM_BATTING_HR * TEAM_BATTING_BB) remains strong and positive
Confirms that HRs are more valuable when combined with walks. Suggests plate discipline (BBs) is crucial for power hitters. Decision: Should we keep TEAM_BATTING_2B?
If VIF test shows high correlation with TEAM_BATTING_H, we should drop it. If interaction terms (e.g., TEAM_BATTING_2B * TEAM_BATTING_H) make sense, we could try that instead.
plot(improved_model, which=1)
Our diagnostic plots show a fairly linear model. ##### Normality Check:
plot(improved_model, which=2)
shapiro.test(residuals(improved_model))
##
## Shapiro-Wilk normality test
##
## data: residuals(improved_model)
## W = 0.99714, p-value = 0.003293
# Shapiro-Wilk normality test: look for high p-value
Our QQ plot suggests normality thought there is obvious skewing on the tails, particularly on the right.
A Shapiro Wilk’s Test statistic had a value of W = 0.9971 suggesting normality.
However, the p-value (0.0033) is less than < 0.05 suggesting that residuals do not follow a normal distribution.
plot(improved_model, which=3)
bptest(improved_model)
##
## studentized Breusch-Pagan test
##
## data: improved_model
## BP = 200.46, df = 7, p-value < 2.2e-16
# Breusch-Pagan test; look for high p-value
Test Statistic (BP = 200.46) suggest higher evidence of heteroscedasticity. Additionally, the p-value is extremely small (much less than 0.05), suggesting higher evidence of heteroscedasticity and that we may need to transform the data to meet the Assumption of Homoscedasticity.
acf(residuals(improved_model))
durbinWatsonTest(improved_model)
## lag Autocorrelation D-W Statistic p-value
## 1 0.002609554 1.991934 0.884
## Alternative hypothesis: rho != 0
# Durbin Watson should be close to 2
Our Autocorrelation Function shows that there are lags above the blue dashed line, suggesting no autocorrelation. This is confirmed through a Durbin-Watson test statistic value of 2.03 and an autocorrelation value of -0.0017. Furthermore, as our p-value (0.558) is greater than 0.05, we do not have enough evidence to reject the null hypothesis that there is no autocorrelation. In other words, the test results suggest that our model’s residuals are independent and therefore do not violate the Independence Assumption.
# Check Variance Inflation Factor (VIF)
vif(improved_model)
## there are higher-order terms (interactions) in this model
## consider setting type = 'predictor'; see ?vif
## TEAM_BATTING_H TEAM_BATTING_HR
## 2.573335 28.363619
## TEAM_BATTING_BB TEAM_BATTING_2B
## 4.310450 2.339192
## TEAM_PITCHING_SO TEAM_FIELDING_E
## 1.701116 2.767363
## TEAM_BATTING_HR:TEAM_BATTING_BB
## 34.483096
TEAM_BATTING_H, TEAM_BATTING_2B, TEAM_PITCHING_SO, TEAM_FIELDING_E are in the range of (1-5) - No significant multicollinearity (good)
TEAM_BATTING_HR, TEAM_BATTING_HR:TEAM_BATTING_BB are in the range of (> 10) shows Severe multicollinearity (highly problematic).
We select variables based on correlation with TARGET_WINS and ensure they are not highly correlated with each other (VIF < 5).
library(car)
# Manually selected high-impact variables
high_impact_model <- lm(TARGET_WINS ~ TEAM_BATTING_H + TEAM_BATTING_2B + TEAM_BATTING_HR +
TEAM_PITCHING_HR + TEAM_PITCHING_SO + TEAM_FIELDING_E, data = stp75_train_df)
# View model summary
summary(high_impact_model)
##
## Call:
## lm(formula = TARGET_WINS ~ TEAM_BATTING_H + TEAM_BATTING_2B +
## TEAM_BATTING_HR + TEAM_PITCHING_HR + TEAM_PITCHING_SO + TEAM_FIELDING_E,
## data = stp75_train_df)
##
## Residuals:
## Min 1Q Median 3Q Max
## -51.719 -9.068 0.023 9.382 53.464
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 8.8569818 4.9380875 1.794 0.07305 .
## TEAM_BATTING_H 0.0584401 0.0039920 14.639 < 2e-16 ***
## TEAM_BATTING_2B -0.0379419 0.0108106 -3.510 0.00046 ***
## TEAM_BATTING_HR -0.0179674 0.0285873 -0.629 0.52975
## TEAM_PITCHING_HR 0.0230521 0.0282249 0.817 0.41420
## TEAM_PITCHING_SO 0.0005344 0.0016008 0.334 0.73854
## TEAM_FIELDING_E -0.0243297 0.0021563 -11.283 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 13.62 on 1697 degrees of freedom
## Multiple R-squared: 0.2357, Adjusted R-squared: 0.233
## F-statistic: 87.24 on 6 and 1697 DF, p-value: < 2.2e-16
plot(high_impact_model)
# Check for multicollinearity
vif(high_impact_model)
## TEAM_BATTING_H TEAM_BATTING_2B TEAM_BATTING_HR TEAM_PITCHING_HR
## 2.793787 2.345585 27.684325 27.666942
## TEAM_PITCHING_SO TEAM_FIELDING_E
## 1.925030 2.087761
TEAM_BATTING_2B (Doubles): Important for offensive production.
TEAM_PITCHING_HR (Home Runs Allowed): Directly impacts opponent scoring (negative impact).
Removed highly correlated variables (VIF > 5).
What changed?
Better feature selection → Based on both domain knowledge and correlation analysis.
Removes redundant variables that cause multicollinearity.
# Apply log transformation to selected variables
train_df_log <- stp75_train_df %>%
mutate(
log_BATTING_H = log1p(TEAM_BATTING_H),
log_BATTING_HR = log1p(TEAM_BATTING_HR),
log_PITCHING_SO = log1p(TEAM_PITCHING_SO)
)
# Fit model with transformed variables
log_model <- lm(TARGET_WINS ~ log_BATTING_H + log_BATTING_HR + log_PITCHING_SO + TEAM_FIELDING_E, data = train_df_log)
# View model summary
summary(log_model)
##
## Call:
## lm(formula = TARGET_WINS ~ log_BATTING_H + log_BATTING_HR + log_PITCHING_SO +
## TEAM_FIELDING_E, data = train_df_log)
##
## Residuals:
## Min 1Q Median 3Q Max
## -46.859 -9.396 0.188 9.407 48.822
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -5.695e+02 3.753e+01 -15.174 < 2e-16 ***
## log_BATTING_H 8.754e+01 4.575e+00 19.136 < 2e-16 ***
## log_BATTING_HR -2.767e+00 5.976e-01 -4.631 3.91e-06 ***
## log_PITCHING_SO 4.619e+00 1.346e+00 3.431 0.000617 ***
## TEAM_FIELDING_E -2.713e-02 2.103e-03 -12.899 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 13.59 on 1699 degrees of freedom
## Multiple R-squared: 0.238, Adjusted R-squared: 0.2362
## F-statistic: 132.6 on 4 and 1699 DF, p-value: < 2.2e-16
plot(log_model)
#### Why log transformation?
Fixes right-skewed distributions (e.g., extreme HR and SO values).
Helps meet the linear regression assumption of normality.
Reduces outlier influence.
What changed?
More stable regression coefficients with reduced variability.
Improves the model fit for non-linear relationships.
summary(base_model)$adj.r.squared
## [1] 0.2279034
summary(high_impact_model)$adj.r.squared
## [1] 0.2330402
summary(log_model)$adj.r.squared
## [1] 0.236158
mse_base <- mean((stp75_train_df$TARGET_WINS - predict(base_model, stp75_train_df))^2)
mse_high_impact <- mean((stp75_train_df$TARGET_WINS - predict(high_impact_model, stp75_train_df))^2)
mse_log <- mean((train_df_log$TARGET_WINS - predict(log_model, train_df_log))^2)
print(c(mse_base, mse_high_impact, mse_log))
## [1] 186.2483 184.7914 184.2571
vif(high_impact_model)
## TEAM_BATTING_H TEAM_BATTING_2B TEAM_BATTING_HR TEAM_PITCHING_HR
## 2.793787 2.345585 27.684325 27.666942
## TEAM_PITCHING_SO TEAM_FIELDING_E
## 1.925030 2.087761
For this model, we are using backward selection to arrive at the smallest number of variables with statistical significance. In this case, we begin by creating a model containing all predictors, then gradually remove the variable with the highest P-value until only the variables with statistical significance remain.
We will use Best Subset Selection to get a estimate of the optimal number of predictors in our model. This test uses Mallows’ Cp.
library(leaps)
regfit_full = regsubsets(TARGET_WINS ~ ., data = stp75_train_df, nvmax = 11)
regfit_summary = summary(regfit_full)
plot(regfit_summary$cp, xlab="Number of variables", ylab="cp")
points(which.min(regfit_summary$cp),regfit_summary$cp[which.min(regfit_summary$cp)], pch=20,col="red")
Based on the Best Subset Selection method, we estimate that our
model should have 9 observations.
As an alternative to Best Subset Selection, we used the Cross Validation method to estimate the optimal number of predictors in our model. Cross Validation divides our training dataset into k - 1 number of “folds”, then tests the data on the kth “fold”. For our test, we used five folds (k = 5).
set.seed(11)
folds=sample(rep(1:5,length=nrow(stp75_train_df)))
cv_errors = matrix(NA,5,10)
for(k in 1:5) {
best_fit = regsubsets(TARGET_WINS ~ ., data=stp75_train_df[folds!=k,], nvmax=10, method="forward")
for(i in 1:10) {
# Extract the selected coefficients for the i-th model
selected_coefs = coef(best_fit, id = i)
# Predict manually by calculating the linear combination of the features
# First, subset the data for the k-th fold
test_data = stp75_train_df[folds == k, ]
# Only include the predictors that were selected
predictors = names(selected_coefs)[-1] # Exclude the intercept term
# Calculate the predictions (including the intercept)
pred = as.matrix(test_data[, predictors]) %*% selected_coefs[predictors] + selected_coefs[1]
cv_errors[k,i]=mean((stp75_train_df$TARGET_WINS[folds==k] - pred)^2)
}
}
rmse_cv = sqrt(apply(cv_errors,2,mean))
plot(rmse_cv, pch=5, type="b")
Based on the Cross Validation method, we estimate that our
model should have 9 observations.
summary(stp_model_full)
##
## Call:
## lm(formula = TARGET_WINS ~ ., data = stp75_train_df)
##
## Residuals:
## Min 1Q Median 3Q Max
## -47.770 -8.578 0.104 8.339 57.289
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 18.4674648 6.2805997 2.940 0.003322 **
## TEAM_BATTING_H 0.0501921 0.0044617 11.250 < 2e-16 ***
## TEAM_BATTING_2B -0.0232963 0.0105809 -2.202 0.027818 *
## TEAM_BATTING_3B 0.0704114 0.0199153 3.536 0.000418 ***
## TEAM_BATTING_HR -0.0143389 0.0352355 -0.407 0.684100
## TEAM_BATTING_BB 0.0167559 0.0073216 2.289 0.022228 *
## TEAM_BATTING_SO -0.0028343 0.0038858 -0.729 0.465858
## TEAM_BASERUN_SB 0.0286960 0.0051909 5.528 3.74e-08 ***
## TEAM_BASERUN_CS -0.0149534 0.0180982 -0.826 0.408787
## TEAM_PITCHING_H -0.0005012 0.0004910 -1.021 0.307539
## TEAM_PITCHING_HR 0.0660800 0.0323533 2.042 0.041262 *
## TEAM_PITCHING_BB -0.0071463 0.0054045 -1.322 0.186252
## TEAM_PITCHING_SO 0.0013541 0.0024422 0.554 0.579346
## TEAM_FIELDING_E -0.0213156 0.0029091 -7.327 3.63e-13 ***
## TEAM_FIELDING_DP -0.1061426 0.0148629 -7.141 1.37e-12 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 13.01 on 1689 degrees of freedom
## Multiple R-squared: 0.3063, Adjusted R-squared: 0.3005
## F-statistic: 53.26 on 14 and 1689 DF, p-value: < 2.2e-16
AIC(stp_model_full)
## [1] 13596.37
TEAM_BATTING_HR has the highest p-value. We removed TEAM_BATTING_HR from our predictors and updated the model.
back_select_model <- update(stp_model_full, . ~ . - TEAM_BATTING_HR)
summary(back_select_model)
##
## Call:
## lm(formula = TARGET_WINS ~ TEAM_BATTING_H + TEAM_BATTING_2B +
## TEAM_BATTING_3B + TEAM_BATTING_BB + TEAM_BATTING_SO + TEAM_BASERUN_SB +
## TEAM_BASERUN_CS + TEAM_PITCHING_H + TEAM_PITCHING_HR + TEAM_PITCHING_BB +
## TEAM_PITCHING_SO + TEAM_FIELDING_E + TEAM_FIELDING_DP, data = stp75_train_df)
##
## Residuals:
## Min 1Q Median 3Q Max
## -47.787 -8.579 0.087 8.351 57.401
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 18.5254526 6.2774327 2.951 0.003210 **
## TEAM_BATTING_H 0.0502361 0.0044592 11.266 < 2e-16 ***
## TEAM_BATTING_2B -0.0233482 0.0105775 -2.207 0.027423 *
## TEAM_BATTING_3B 0.0712736 0.0197974 3.600 0.000327 ***
## TEAM_BATTING_BB 0.0156323 0.0067793 2.306 0.021238 *
## TEAM_BATTING_SO -0.0032609 0.0037408 -0.872 0.383494
## TEAM_BASERUN_SB 0.0287248 0.0051892 5.536 3.59e-08 ***
## TEAM_BASERUN_CS -0.0143034 0.0180232 -0.794 0.427532
## TEAM_PITCHING_H -0.0005316 0.0004851 -1.096 0.273307
## TEAM_PITCHING_HR 0.0535904 0.0102340 5.236 1.84e-07 ***
## TEAM_PITCHING_BB -0.0062913 0.0049782 -1.264 0.206485
## TEAM_PITCHING_SO 0.0015803 0.0023776 0.665 0.506359
## TEAM_FIELDING_E -0.0213046 0.0029083 -7.326 3.67e-13 ***
## TEAM_FIELDING_DP -0.1063550 0.0148500 -7.162 1.18e-12 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 13.01 on 1690 degrees of freedom
## Multiple R-squared: 0.3062, Adjusted R-squared: 0.3008
## F-statistic: 57.37 on 13 and 1690 DF, p-value: < 2.2e-16
AIC(back_select_model)
## [1] 13594.53
TEAM_PITCHING_SO has the highest p-value. We removed TEAM_PITCHING_SO from our predictors and update the model.
back_select_model <- update(back_select_model, . ~ . - TEAM_PITCHING_SO)
summary(back_select_model)
##
## Call:
## lm(formula = TARGET_WINS ~ TEAM_BATTING_H + TEAM_BATTING_2B +
## TEAM_BATTING_3B + TEAM_BATTING_BB + TEAM_BATTING_SO + TEAM_BASERUN_SB +
## TEAM_BASERUN_CS + TEAM_PITCHING_H + TEAM_PITCHING_HR + TEAM_PITCHING_BB +
## TEAM_FIELDING_E + TEAM_FIELDING_DP, data = stp75_train_df)
##
## Residuals:
## Min 1Q Median 3Q Max
## -47.829 -8.630 0.114 8.304 57.451
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 18.7441482 6.2677688 2.991 0.002825 **
## TEAM_BATTING_H 0.0500658 0.0044511 11.248 < 2e-16 ***
## TEAM_BATTING_2B -0.0229972 0.0105625 -2.177 0.029601 *
## TEAM_BATTING_3B 0.0720801 0.0197569 3.648 0.000272 ***
## TEAM_BATTING_BB 0.0133025 0.0058018 2.293 0.021981 *
## TEAM_BATTING_SO -0.0014733 0.0025996 -0.567 0.570976
## TEAM_BASERUN_SB 0.0283374 0.0051555 5.497 4.46e-08 ***
## TEAM_BASERUN_CS -0.0139882 0.0180140 -0.777 0.437549
## TEAM_PITCHING_H -0.0005686 0.0004819 -1.180 0.238199
## TEAM_PITCHING_HR 0.0536376 0.0102321 5.242 1.79e-07 ***
## TEAM_PITCHING_BB -0.0043455 0.0040258 -1.079 0.280550
## TEAM_FIELDING_E -0.0209759 0.0028655 -7.320 3.81e-13 ***
## TEAM_FIELDING_DP -0.1065292 0.0148453 -7.176 1.07e-12 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 13 on 1691 degrees of freedom
## Multiple R-squared: 0.306, Adjusted R-squared: 0.3011
## F-statistic: 62.13 on 12 and 1691 DF, p-value: < 2.2e-16
AIC(back_select_model)
## [1] 13592.98
TEAM_BATTING_SO has the highest p-value. We removed TEAM_BATTING_SO from our predictors and update the model.
back_select_model <- update(back_select_model, . ~ . - TEAM_BATTING_SO)
summary(back_select_model)
##
## Call:
## lm(formula = TARGET_WINS ~ TEAM_BATTING_H + TEAM_BATTING_2B +
## TEAM_BATTING_3B + TEAM_BATTING_BB + TEAM_BASERUN_SB + TEAM_BASERUN_CS +
## TEAM_PITCHING_H + TEAM_PITCHING_HR + TEAM_PITCHING_BB + TEAM_FIELDING_E +
## TEAM_FIELDING_DP, data = stp75_train_df)
##
## Residuals:
## Min 1Q Median 3Q Max
## -47.727 -8.628 0.086 8.333 57.129
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 16.1571222 4.2941305 3.763 0.000174 ***
## TEAM_BATTING_H 0.0513371 0.0038438 13.356 < 2e-16 ***
## TEAM_BATTING_2B -0.0246716 0.0101389 -2.433 0.015062 *
## TEAM_BATTING_3B 0.0744296 0.0193132 3.854 0.000121 ***
## TEAM_BATTING_BB 0.0133550 0.0057999 2.303 0.021420 *
## TEAM_BASERUN_SB 0.0274615 0.0049174 5.585 2.73e-08 ***
## TEAM_BASERUN_CS -0.0137429 0.0180051 -0.763 0.445404
## TEAM_PITCHING_H -0.0005572 0.0004814 -1.158 0.247196
## TEAM_PITCHING_HR 0.0503000 0.0083657 6.013 2.23e-09 ***
## TEAM_PITCHING_BB -0.0039640 0.0039683 -0.999 0.317976
## TEAM_FIELDING_E -0.0209665 0.0028648 -7.319 3.85e-13 ***
## TEAM_FIELDING_DP -0.1058938 0.0147999 -7.155 1.24e-12 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 13 on 1692 degrees of freedom
## Multiple R-squared: 0.3059, Adjusted R-squared: 0.3014
## F-statistic: 67.78 on 11 and 1692 DF, p-value: < 2.2e-16
AIC(back_select_model)
## [1] 13591.3
TEAM_BASERUN_CS has the highest p-value. We removed TEAM_BASERUN_CS from our predictors and update the model.
back_select_model <- update(back_select_model, . ~ . - TEAM_BASERUN_CS)
summary(back_select_model)
##
## Call:
## lm(formula = TARGET_WINS ~ TEAM_BATTING_H + TEAM_BATTING_2B +
## TEAM_BATTING_3B + TEAM_BATTING_BB + TEAM_BASERUN_SB + TEAM_PITCHING_H +
## TEAM_PITCHING_HR + TEAM_PITCHING_BB + TEAM_FIELDING_E + TEAM_FIELDING_DP,
## data = stp75_train_df)
##
## Residuals:
## Min 1Q Median 3Q Max
## -47.649 -8.602 0.012 8.342 57.080
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 15.3222848 4.1519888 3.690 0.000231 ***
## TEAM_BATTING_H 0.0513101 0.0038431 13.351 < 2e-16 ***
## TEAM_BATTING_2B -0.0249908 0.0101290 -2.467 0.013714 *
## TEAM_BATTING_3B 0.0744461 0.0193108 3.855 0.000120 ***
## TEAM_BATTING_BB 0.0136940 0.0057821 2.368 0.017981 *
## TEAM_BASERUN_SB 0.0265482 0.0047690 5.567 3.01e-08 ***
## TEAM_PITCHING_H -0.0005841 0.0004800 -1.217 0.223869
## TEAM_PITCHING_HR 0.0517755 0.0081382 6.362 2.56e-10 ***
## TEAM_PITCHING_BB -0.0039681 0.0039678 -1.000 0.317412
## TEAM_FIELDING_E -0.0204796 0.0027926 -7.334 3.46e-13 ***
## TEAM_FIELDING_DP -0.1063293 0.0147871 -7.191 9.63e-13 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 13 on 1693 degrees of freedom
## Multiple R-squared: 0.3056, Adjusted R-squared: 0.3015
## F-statistic: 74.52 on 10 and 1693 DF, p-value: < 2.2e-16
AIC(back_select_model)
## [1] 13589.89
TEAM_PITCHING_BB has the highest p-value. We removed TEAM_PITCHING_BB from our predictors and update the model.
back_select_model <- update(back_select_model, . ~ . - TEAM_PITCHING_BB)
summary(back_select_model)
##
## Call:
## lm(formula = TARGET_WINS ~ TEAM_BATTING_H + TEAM_BATTING_2B +
## TEAM_BATTING_3B + TEAM_BATTING_BB + TEAM_BASERUN_SB + TEAM_PITCHING_H +
## TEAM_PITCHING_HR + TEAM_FIELDING_E + TEAM_FIELDING_DP, data = stp75_train_df)
##
## Residuals:
## Min 1Q Median 3Q Max
## -48.340 -8.550 -0.023 8.399 56.895
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 15.2339468 4.1510493 3.670 0.000250 ***
## TEAM_BATTING_H 0.0521105 0.0037589 13.863 < 2e-16 ***
## TEAM_BATTING_2B -0.0253972 0.0101209 -2.509 0.012186 *
## TEAM_BATTING_3B 0.0709375 0.0189894 3.736 0.000193 ***
## TEAM_BATTING_BB 0.0093133 0.0037744 2.468 0.013704 *
## TEAM_BASERUN_SB 0.0258288 0.0047145 5.479 4.93e-08 ***
## TEAM_PITCHING_H -0.0008524 0.0003980 -2.141 0.032388 *
## TEAM_PITCHING_HR 0.0506989 0.0080667 6.285 4.16e-10 ***
## TEAM_FIELDING_E -0.0208150 0.0027724 -7.508 9.65e-14 ***
## TEAM_FIELDING_DP -0.1065033 0.0147861 -7.203 8.83e-13 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 13 on 1694 degrees of freedom
## Multiple R-squared: 0.3052, Adjusted R-squared: 0.3015
## F-statistic: 82.69 on 9 and 1694 DF, p-value: < 2.2e-16
AIC(back_select_model)
## [1] 13588.9
Backward selection using P-values arrives at a model with six variables with high statistical significance ((p-value < 0.001) and three with moderate (p-value ~= 0.1) statistical significance. Arriving at 11 predictors is consistent with our Best Subset Selection test.
vif(back_select_model)
## TEAM_BATTING_H TEAM_BATTING_2B TEAM_BATTING_3B TEAM_BATTING_BB
## 2.719827 2.257416 2.673757 2.133184
## TEAM_BASERUN_SB TEAM_PITCHING_H TEAM_PITCHING_HR TEAM_FIELDING_E
## 1.587993 2.329005 2.481519 3.789448
## TEAM_FIELDING_DP
## 1.339361
plot(back_select_model)
Another approach to model selection involves a combination of step-wise selection and contextual selection. In this example, we can first drop TEAM_BATTING_2B (2 base hits), TEAM_BATTING_3B (3 base hits), TEAM_BATTING_HR (home runs) before conducting our stepwise selection, as data for these columns is represented in the TEAM_BATTING_H (total hits) column, thus signaling the existence of dependence between hit count variables and total hits. It is not totally clear if TEAM_PITCHING_HR is also included in TEAM_PITCHING_H from the supporting document but we will remove it in the event that a similar relationship exists. Alternatively, we could have created a new column (TEAM_BATTING_1B) by subtracting the sum of 2 base hits, 3 base hits, and homeruns from total hits, but this alternatively solution would have a) added complexity into our model and b) added uncertainty due to possible missing values in the home runs columns.
stp75s_train_df <- stp75_train_df |>
drop_columns(c("TEAM_BATTING_2B", "TEAM_BATTING_3B", "TEAM_BATTING_HR", "TEAM_PITCHING_HR"))
stp25s_test_df <- stp25_test_df |>
drop_columns(c("TEAM_BATTING_2B", "TEAM_BATTING_3B", "TEAM_BATTING_HR", "TEAM_PITCHING_HR"))
stp_model_sm <- lm(TARGET_WINS ~ ., data = stp75s_train_df)
# Backward step-wise regression
stpb_model_sm <- step(stp_model_sm, direction = "backward")
## Start: AIC=8789.49
## TARGET_WINS ~ TEAM_BATTING_H + TEAM_BATTING_BB + TEAM_BATTING_SO +
## TEAM_BASERUN_SB + TEAM_BASERUN_CS + TEAM_PITCHING_H + TEAM_PITCHING_BB +
## TEAM_PITCHING_SO + TEAM_FIELDING_E + TEAM_FIELDING_DP
##
## Df Sum of Sq RSS AIC
## - TEAM_BATTING_SO 1 1 292429 8787.5
## - TEAM_PITCHING_BB 1 33 292461 8787.7
## - TEAM_PITCHING_SO 1 91 292519 8788.0
## - TEAM_PITCHING_H 1 269 292697 8789.1
## <none> 292428 8789.5
## - TEAM_BASERUN_CS 1 525 292954 8790.5
## - TEAM_BATTING_BB 1 909 293338 8792.8
## - TEAM_BASERUN_SB 1 6102 298530 8822.7
## - TEAM_FIELDING_DP 1 7670 300099 8831.6
## - TEAM_FIELDING_E 1 9439 301868 8841.6
## - TEAM_BATTING_H 1 66689 359117 9137.5
##
## Step: AIC=8787.5
## TARGET_WINS ~ TEAM_BATTING_H + TEAM_BATTING_BB + TEAM_BASERUN_SB +
## TEAM_BASERUN_CS + TEAM_PITCHING_H + TEAM_PITCHING_BB + TEAM_PITCHING_SO +
## TEAM_FIELDING_E + TEAM_FIELDING_DP
##
## Df Sum of Sq RSS AIC
## - TEAM_PITCHING_BB 1 57 292487 8785.8
## - TEAM_PITCHING_H 1 272 292701 8787.1
## <none> 292429 8787.5
## - TEAM_PITCHING_SO 1 367 292797 8787.6
## - TEAM_BASERUN_CS 1 541 292970 8788.6
## - TEAM_BATTING_BB 1 1194 293623 8792.4
## - TEAM_BASERUN_SB 1 6215 298644 8821.3
## - TEAM_FIELDING_DP 1 7692 300122 8829.7
## - TEAM_FIELDING_E 1 10473 302903 8845.5
## - TEAM_BATTING_H 1 69454 361883 9148.6
##
## Step: AIC=8785.83
## TARGET_WINS ~ TEAM_BATTING_H + TEAM_BATTING_BB + TEAM_BASERUN_SB +
## TEAM_BASERUN_CS + TEAM_PITCHING_H + TEAM_PITCHING_SO + TEAM_FIELDING_E +
## TEAM_FIELDING_DP
##
## Df Sum of Sq RSS AIC
## - TEAM_PITCHING_SO 1 313 292799 8785.7
## <none> 292487 8785.8
## - TEAM_BASERUN_CS 1 553 293040 8787.1
## - TEAM_PITCHING_H 1 573 293059 8787.2
## - TEAM_BATTING_BB 1 2122 294609 8796.2
## - TEAM_BASERUN_SB 1 6195 298682 8819.5
## - TEAM_FIELDING_DP 1 7687 300173 8828.0
## - TEAM_FIELDING_E 1 11084 303571 8847.2
## - TEAM_BATTING_H 1 69638 362125 9147.8
##
## Step: AIC=8785.65
## TARGET_WINS ~ TEAM_BATTING_H + TEAM_BATTING_BB + TEAM_BASERUN_SB +
## TEAM_BASERUN_CS + TEAM_PITCHING_H + TEAM_FIELDING_E + TEAM_FIELDING_DP
##
## Df Sum of Sq RSS AIC
## <none> 292799 8785.7
## - TEAM_PITCHING_H 1 410 293209 8786.0
## - TEAM_BASERUN_CS 1 694 293494 8787.7
## - TEAM_BATTING_BB 1 2042 294842 8795.5
## - TEAM_BASERUN_SB 1 6320 299119 8820.0
## - TEAM_FIELDING_DP 1 7515 300315 8826.8
## - TEAM_FIELDING_E 1 11574 304373 8849.7
## - TEAM_BATTING_H 1 77679 370478 9184.6
Performing an automated step-wise backward selection gives us 8 viable predictors. Examining the VIF shows that eight parameters are not strongly correlated with our dependent variable.
vif(stpb_model_sm)
## TEAM_BATTING_H TEAM_BATTING_BB TEAM_BASERUN_SB TEAM_BASERUN_CS
## 1.207011 2.074158 1.566007 1.132796
## TEAM_PITCHING_H TEAM_FIELDING_E TEAM_FIELDING_DP
## 2.143770 3.396556 1.291391
summary(stpb_model_sm)
##
## Call:
## lm(formula = TARGET_WINS ~ TEAM_BATTING_H + TEAM_BATTING_BB +
## TEAM_BASERUN_SB + TEAM_BASERUN_CS + TEAM_PITCHING_H + TEAM_FIELDING_E +
## TEAM_FIELDING_DP, data = stp75s_train_df)
##
## Residuals:
## Min 1Q Median 3Q Max
## -47.396 -8.750 0.183 8.431 52.491
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 14.1313943 4.1032804 3.444 0.000587 ***
## TEAM_BATTING_H 0.0536872 0.0025310 21.212 < 2e-16 ***
## TEAM_BATTING_BB 0.0129380 0.0037619 3.439 0.000597 ***
## TEAM_BASERUN_SB 0.0286310 0.0047321 6.050 1.77e-09 ***
## TEAM_BASERUN_CS -0.0351827 0.0175441 -2.005 0.045081 *
## TEAM_PITCHING_H -0.0005947 0.0003860 -1.541 0.123606
## TEAM_FIELDING_E -0.0217219 0.0026530 -8.188 5.17e-16 ***
## TEAM_FIELDING_DP -0.0968235 0.0146751 -6.598 5.56e-11 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 13.14 on 1696 degrees of freedom
## Multiple R-squared: 0.2893, Adjusted R-squared: 0.2864
## F-statistic: 98.65 on 7 and 1696 DF, p-value: < 2.2e-16
AIC(stpb_model_sm)
## [1] 13623.39
plot(stpb_model_sm)
Looking at the p-values and t-values for the remaining 7 predictors show that there is one variables without statistical significance, TEAM_PITCHING_H. As this predictor has a high p-value (0.12) and a low t-value (-1.54), we should consider removing it.
stpb_model_xs <- update(stpb_model_sm, . ~ . - TEAM_PITCHING_H)
summary(stpb_model_xs)
##
## Call:
## lm(formula = TARGET_WINS ~ TEAM_BATTING_H + TEAM_BATTING_BB +
## TEAM_BASERUN_SB + TEAM_BASERUN_CS + TEAM_FIELDING_E + TEAM_FIELDING_DP,
## data = stp75s_train_df)
##
## Residuals:
## Min 1Q Median 3Q Max
## -47.070 -8.834 0.110 8.350 53.599
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 15.111949 4.055255 3.727 0.000201 ***
## TEAM_BATTING_H 0.052652 0.002441 21.569 < 2e-16 ***
## TEAM_BATTING_BB 0.012851 0.003763 3.415 0.000653 ***
## TEAM_BASERUN_SB 0.030592 0.004559 6.710 2.65e-11 ***
## TEAM_BASERUN_CS -0.035804 0.017547 -2.040 0.041456 *
## TEAM_FIELDING_E -0.024057 0.002178 -11.045 < 2e-16 ***
## TEAM_FIELDING_DP -0.097476 0.014675 -6.642 4.15e-11 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 13.14 on 1697 degrees of freedom
## Multiple R-squared: 0.2884, Adjusted R-squared: 0.2858
## F-statistic: 114.6 on 6 and 1697 DF, p-value: < 2.2e-16
AIC(stpb_model_xs)
## [1] 13623.78
plot(stpb_model_xs)
Removing the TEAM_PITCHING_H slightly increases our AIC from 13623.39 to 13623.78 and slightly decreases the Adjusted R-squared from 0.2864 to 0.2858, but overall the models are somewhat similar the auto-generated step-wise selected model.
The function compare_performance allows comparison of various metrics across diffrent models including R2 and R2 (adj.). It also weights AIC and BIC values to help rank the performance of our models. Based on the result of this test, it appears that our original backward-selected model performed best.
compare_performance(back_select_model, stpb_model_sm, stpb_model_xs, rank = TRUE)
In this section, we will test if our original backward-selected model meets the four assumptions for linearity.
plot(back_select_model, which=1)
Our diagnostic plots show a fairly linear model.
plot(back_select_model, which=2)
shapiro.test(residuals(back_select_model))
##
## Shapiro-Wilk normality test
##
## data: residuals(back_select_model)
## W = 0.9967, p-value = 0.001034
# Shapiro-Wilk normality test: look for high p-value
Our QQ plot suggests normality thought there is obvious skewing on the tails, particularly on the right.
A Shapiro Wilk’s Test statistic had a value of 0.9967 is close to 1 suggesting normality. However, the p-value (0.0010) is less than < 0.05 suggests that residuals may not follow a normal distribution.
plot(back_select_model, which=3)
plot(back_select_model, which=4)
bptest(back_select_model)
##
## studentized Breusch-Pagan test
##
## data: back_select_model
## BP = 210.37, df = 9, p-value < 2.2e-16
# Breusch-Pagan test; look for high p-value
Our Scale-Location plot shows that points appear somewhat evenly distributed above and below the trend line. While there is no obvious fan/wedge pattern, there is clustering in the center suggesting underfitting, high leverage outliers or that additional transformation may be needed. As our Cook’s Distance plot has no values with a greater than 1, we can rule our the effects of high leverage points.
The Breusch-Page test statistic BP (210.37) and the small p-value (2.2e-16) suggest evidence of heteroscedasticity. Though the values are different that we may need to transform the data to meet the Assumption of Homoscedasticity.
acf(residuals(back_select_model))
durbinWatsonTest(back_select_model)
## lag Autocorrelation D-W Statistic p-value
## 1 -0.01715261 2.028732 0.59
## Alternative hypothesis: rho != 0
# Durbin Watson should be close to 2
Our Autocorrelation Function shows that there are lags above the blue dashed line, suggesting no autocorrelation. This is confirmed through a Durbin-Watson test statistic value of 2.03 and an autocorrelation value of -0.0017. Furthermore, as our p-value (0.558) is greater than 0.05, we do not have enough evidence to reject the null hypothesis that there is no autocorrelation. In other words, the test results suggest that our model’s residuals are independent and therefore do not violate the Independence Assumption.
Conclusion The backward selected model violates the Homoscedasticity Assumption and has mixed results for our Normality Assumption. We should therefore test applying transformations to our model.
So far, our diagnostic plots show that our model is somewhat linear, independent, heteroschedastic, but our residuals vs leverage indicates the presence of outliers. We will attempt to use transformations to improve our model.
In this section, we apply a Box-Cox Transformation
stpbc_model <- boxcox(back_select_model, lambda = seq(-3,3))
plot(stpbc_model)
best_lambda <- stpbc_model$x[which(stpbc_model$y==max(stpbc_model$y))]
stp_model_inv <- lm((TARGET_WINS)^best_lambda ~ TEAM_BATTING_H + TEAM_BATTING_2B + TEAM_BATTING_3B + TEAM_BATTING_SO + TEAM_BASERUN_SB + TEAM_PITCHING_HR + TEAM_PITCHING_BB + TEAM_PITCHING_SO + TEAM_FIELDING_E + TEAM_FIELDING_DP, data = stp75_train_df)
summary(stp_model_inv)
##
## Call:
## lm(formula = (TARGET_WINS)^best_lambda ~ TEAM_BATTING_H + TEAM_BATTING_2B +
## TEAM_BATTING_3B + TEAM_BATTING_SO + TEAM_BASERUN_SB + TEAM_PITCHING_HR +
## TEAM_PITCHING_BB + TEAM_PITCHING_SO + TEAM_FIELDING_E + TEAM_FIELDING_DP,
## data = stp75_train_df)
##
## Residuals:
## Min 1Q Median 3Q Max
## -181.370 -31.826 -1.085 29.386 220.329
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 24.328584 21.671396 1.123 0.2618
## TEAM_BATTING_H 0.172877 0.015518 11.140 < 2e-16 ***
## TEAM_BATTING_2B -0.077426 0.037841 -2.046 0.0409 *
## TEAM_BATTING_3B 0.283539 0.069167 4.099 4.34e-05 ***
## TEAM_BATTING_SO 0.002826 0.012382 0.228 0.8195
## TEAM_BASERUN_SB 0.106732 0.017584 6.070 1.58e-09 ***
## TEAM_PITCHING_HR 0.200389 0.036068 5.556 3.20e-08 ***
## TEAM_PITCHING_BB 0.005050 0.009782 0.516 0.6057
## TEAM_PITCHING_SO -0.007150 0.007208 -0.992 0.3213
## TEAM_FIELDING_E -0.087885 0.007809 -11.254 < 2e-16 ***
## TEAM_FIELDING_DP -0.356033 0.052671 -6.760 1.90e-11 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 46.64 on 1693 degrees of freedom
## Multiple R-squared: 0.2935, Adjusted R-squared: 0.2894
## F-statistic: 70.34 on 10 and 1693 DF, p-value: < 2.2e-16
Looking at the summary of our new Box-Cox transformed model shows three variables are not statistically significant. We should therefore remove them one at a time.
stp_model_inv <- update(stp_model_inv, . ~ . - TEAM_BATTING_SO)
summary(stp_model_inv)
##
## Call:
## lm(formula = (TARGET_WINS)^best_lambda ~ TEAM_BATTING_H + TEAM_BATTING_2B +
## TEAM_BATTING_3B + TEAM_BASERUN_SB + TEAM_PITCHING_HR + TEAM_PITCHING_BB +
## TEAM_PITCHING_SO + TEAM_FIELDING_E + TEAM_FIELDING_DP, data = stp75_train_df)
##
## Residuals:
## Min 1Q Median 3Q Max
## -181.221 -31.839 -1.126 29.483 220.649
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 27.262063 17.442915 1.563 0.1183
## TEAM_BATTING_H 0.171668 0.014583 11.772 < 2e-16 ***
## TEAM_BATTING_2B -0.075855 0.037198 -2.039 0.0416 *
## TEAM_BATTING_3B 0.280440 0.067802 4.136 3.70e-05 ***
## TEAM_BASERUN_SB 0.108141 0.016459 6.570 6.67e-11 ***
## TEAM_PITCHING_HR 0.204117 0.032149 6.349 2.78e-10 ***
## TEAM_PITCHING_BB 0.004005 0.008641 0.464 0.6431
## TEAM_PITCHING_SO -0.006066 0.005419 -1.119 0.2632
## TEAM_FIELDING_E -0.088581 0.007188 -12.324 < 2e-16 ***
## TEAM_FIELDING_DP -0.355896 0.052652 -6.759 1.90e-11 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 46.63 on 1694 degrees of freedom
## Multiple R-squared: 0.2935, Adjusted R-squared: 0.2898
## F-statistic: 78.2 on 9 and 1694 DF, p-value: < 2.2e-16
AIC(back_select_model)
## [1] 13588.9
stp_model_inv <- update(stp_model_inv, . ~ . - TEAM_PITCHING_BB)
summary(stp_model_inv)
##
## Call:
## lm(formula = (TARGET_WINS)^best_lambda ~ TEAM_BATTING_H + TEAM_BATTING_2B +
## TEAM_BATTING_3B + TEAM_BASERUN_SB + TEAM_PITCHING_HR + TEAM_PITCHING_SO +
## TEAM_FIELDING_E + TEAM_FIELDING_DP, data = stp75_train_df)
##
## Residuals:
## Min 1Q Median 3Q Max
## -177.826 -31.742 -1.087 29.758 220.026
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 27.675276 17.416083 1.589 0.1122
## TEAM_BATTING_H 0.171840 0.014574 11.791 < 2e-16 ***
## TEAM_BATTING_2B -0.076072 0.037187 -2.046 0.0409 *
## TEAM_BATTING_3B 0.284474 0.067225 4.232 2.44e-05 ***
## TEAM_BASERUN_SB 0.109793 0.016065 6.834 1.15e-11 ***
## TEAM_PITCHING_HR 0.207167 0.031461 6.585 6.05e-11 ***
## TEAM_PITCHING_SO -0.005744 0.005373 -1.069 0.2852
## TEAM_FIELDING_E -0.088788 0.007172 -12.380 < 2e-16 ***
## TEAM_FIELDING_DP -0.351605 0.051820 -6.785 1.60e-11 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 46.62 on 1695 degrees of freedom
## Multiple R-squared: 0.2934, Adjusted R-squared: 0.2901
## F-statistic: 87.98 on 8 and 1695 DF, p-value: < 2.2e-16
AIC(back_select_model)
## [1] 13588.9
stp_model_inv <- update(stp_model_inv, . ~ . - TEAM_PITCHING_SO)
summary(stp_model_inv)
##
## Call:
## lm(formula = (TARGET_WINS)^best_lambda ~ TEAM_BATTING_H + TEAM_BATTING_2B +
## TEAM_BATTING_3B + TEAM_BASERUN_SB + TEAM_PITCHING_HR + TEAM_FIELDING_E +
## TEAM_FIELDING_DP, data = stp75_train_df)
##
## Residuals:
## Min 1Q Median 3Q Max
## -178.015 -31.558 -1.138 29.634 218.413
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 16.581549 13.987093 1.185 0.2360
## TEAM_BATTING_H 0.178723 0.013076 13.668 < 2e-16 ***
## TEAM_BATTING_2B -0.084945 0.036250 -2.343 0.0192 *
## TEAM_BATTING_3B 0.289749 0.067047 4.322 1.64e-05 ***
## TEAM_BASERUN_SB 0.107717 0.015948 6.754 1.97e-11 ***
## TEAM_PITCHING_HR 0.191652 0.027914 6.866 9.25e-12 ***
## TEAM_FIELDING_E -0.092108 0.006465 -14.246 < 2e-16 ***
## TEAM_FIELDING_DP -0.345677 0.051525 -6.709 2.66e-11 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 46.62 on 1696 degrees of freedom
## Multiple R-squared: 0.2929, Adjusted R-squared: 0.29
## F-statistic: 100.4 on 7 and 1696 DF, p-value: < 2.2e-16
AIC(back_select_model)
## [1] 13588.9
plot(stp_model_inv, which=1)
Our Residuals vs. Fitted plot suggest a fairly linear model.
plot(stp_model_inv, which = 2)
shapiro.test(residuals(stp_model_inv))
##
## Shapiro-Wilk normality test
##
## data: residuals(stp_model_inv)
## W = 0.99561, p-value = 7.132e-05
# Shapiro-Wilk normality test: look for high p-value
Our QQ plot suggests normality thought there is obvious skewing on the tails, particularly on the right.
A Shapiro Wilk’s Test statistic had a value of 0.996, also suggesting normality. However, since the p-value (7.132e-05) is less than < 0.05 we may still be violating our normality assumption.
plot(stp_model_inv, which = 3)
plot(stp_model_inv, which = 4)
bptest(stp_model_inv)
##
## studentized Breusch-Pagan test
##
## data: stp_model_inv
## BP = 176.76, df = 7, p-value < 2.2e-16
# Breusch-Pagan test; look for high p-value
Our Scale-Location plot shows that points appear somewhat evenly distributed above and below the trend line. While there is no obvious fan/wedge pattern, there is clustering in the center suggesting underfitting, high leverage outliers or that additional transformation may be needed. As our Cook’s Distance plot has no values with a greater than 1, we can rule our the effects of high leverage points.
The Breusch-Page test statistic BP (176.76) and the small p-value (2.2e-16) suggest evidence of heteroscedasticity. Though the values are different that we may need to transform the data to meet the Assumption of Homoscedasticity.
acf(residuals(stp_model_inv))
durbinWatsonTest(stp_model_inv)
## lag Autocorrelation D-W Statistic p-value
## 1 -0.02017478 2.035401 0.42
## Alternative hypothesis: rho != 0
# Durbin Watson should be close to 2
Our Autocorrelation Function shows that there are lags above the blue dashed line, suggesting no autocorrelation. This is confirmed through a Durbin-Watson test statistic value of 2.03 and an autocorrelation value of -0.021. Furthermore, as our p-value (0.416) is greater than 0.05, we do not have enough evidence to reject the null hypothesis that there is no autocorrelation. In other words, the test results suggest that our model’s residuals are independent and therefore do not violate the Independence Assumption.
Conclusion Applying a Box-Cox transformation to our model appears does not seem to have improved the results of our backward selected model. This model continues to violate the Homoscedasticity Assumption and has mixed results for our Normality Assumption.
stp_logy_model <- lm(log(TARGET_WINS + 1) ~., data = stp75_train_df)
# Backward step-wise regression
stpb_logy_model <- step(stp_logy_model, direction = "backward")
## Start: AIC=-5941.6
## log(TARGET_WINS + 1) ~ TEAM_BATTING_H + TEAM_BATTING_2B + TEAM_BATTING_3B +
## TEAM_BATTING_HR + TEAM_BATTING_BB + TEAM_BATTING_SO + TEAM_BASERUN_SB +
## TEAM_BASERUN_CS + TEAM_PITCHING_H + TEAM_PITCHING_HR + TEAM_PITCHING_BB +
## TEAM_PITCHING_SO + TEAM_FIELDING_E + TEAM_FIELDING_DP
##
## Df Sum of Sq RSS AIC
## - TEAM_PITCHING_SO 1 0.0024 51.230 -5943.5
## - TEAM_PITCHING_BB 1 0.0047 51.232 -5943.4
## - TEAM_BATTING_SO 1 0.0084 51.236 -5943.3
## - TEAM_BASERUN_CS 1 0.0139 51.241 -5943.1
## - TEAM_BATTING_BB 1 0.0471 51.275 -5942.0
## - TEAM_BATTING_HR 1 0.0490 51.276 -5942.0
## <none> 51.227 -5941.6
## - TEAM_PITCHING_H 1 0.1907 51.418 -5937.3
## - TEAM_BATTING_2B 1 0.1920 51.419 -5937.2
## - TEAM_PITCHING_HR 1 0.1931 51.421 -5937.2
## - TEAM_BATTING_3B 1 0.3472 51.575 -5932.1
## - TEAM_BASERUN_SB 1 0.9718 52.199 -5911.6
## - TEAM_FIELDING_DP 1 1.4521 52.680 -5896.0
## - TEAM_FIELDING_E 1 3.1596 54.387 -5841.6
## - TEAM_BATTING_H 1 4.4396 55.667 -5802.0
##
## Step: AIC=-5943.51
## log(TARGET_WINS + 1) ~ TEAM_BATTING_H + TEAM_BATTING_2B + TEAM_BATTING_3B +
## TEAM_BATTING_HR + TEAM_BATTING_BB + TEAM_BATTING_SO + TEAM_BASERUN_SB +
## TEAM_BASERUN_CS + TEAM_PITCHING_H + TEAM_PITCHING_HR + TEAM_PITCHING_BB +
## TEAM_FIELDING_E + TEAM_FIELDING_DP
##
## Df Sum of Sq RSS AIC
## - TEAM_BATTING_SO 1 0.0067 51.237 -5945.3
## - TEAM_PITCHING_BB 1 0.0100 51.240 -5945.2
## - TEAM_BASERUN_CS 1 0.0140 51.244 -5945.0
## - TEAM_BATTING_HR 1 0.0466 51.276 -5944.0
## <none> 51.230 -5943.5
## - TEAM_BATTING_BB 1 0.0650 51.295 -5943.4
## - TEAM_PITCHING_H 1 0.1886 51.419 -5939.3
## - TEAM_PITCHING_HR 1 0.1929 51.423 -5939.1
## - TEAM_BATTING_2B 1 0.1947 51.425 -5939.0
## - TEAM_BATTING_3B 1 0.3456 51.576 -5934.1
## - TEAM_BASERUN_SB 1 0.9954 52.225 -5912.7
## - TEAM_FIELDING_DP 1 1.4512 52.681 -5897.9
## - TEAM_FIELDING_E 1 3.2759 54.506 -5839.9
## - TEAM_BATTING_H 1 4.4693 55.699 -5803.0
##
## Step: AIC=-5945.29
## log(TARGET_WINS + 1) ~ TEAM_BATTING_H + TEAM_BATTING_2B + TEAM_BATTING_3B +
## TEAM_BATTING_HR + TEAM_BATTING_BB + TEAM_BASERUN_SB + TEAM_BASERUN_CS +
## TEAM_PITCHING_H + TEAM_PITCHING_HR + TEAM_PITCHING_BB + TEAM_FIELDING_E +
## TEAM_FIELDING_DP
##
## Df Sum of Sq RSS AIC
## - TEAM_PITCHING_BB 1 0.0109 51.247 -5946.9
## - TEAM_BASERUN_CS 1 0.0141 51.251 -5946.8
## - TEAM_BATTING_HR 1 0.0423 51.279 -5945.9
## <none> 51.237 -5945.3
## - TEAM_BATTING_BB 1 0.0617 51.298 -5945.2
## - TEAM_BATTING_2B 1 0.1898 51.426 -5941.0
## - TEAM_PITCHING_H 1 0.1943 51.431 -5940.8
## - TEAM_PITCHING_HR 1 0.1959 51.432 -5940.8
## - TEAM_BATTING_3B 1 0.3397 51.576 -5936.0
## - TEAM_BASERUN_SB 1 1.1429 52.379 -5909.7
## - TEAM_FIELDING_DP 1 1.4766 52.713 -5898.9
## - TEAM_FIELDING_E 1 3.2753 54.512 -5841.7
## - TEAM_BATTING_H 1 5.7039 56.941 -5767.4
##
## Step: AIC=-5946.93
## log(TARGET_WINS + 1) ~ TEAM_BATTING_H + TEAM_BATTING_2B + TEAM_BATTING_3B +
## TEAM_BATTING_HR + TEAM_BATTING_BB + TEAM_BASERUN_SB + TEAM_BASERUN_CS +
## TEAM_PITCHING_H + TEAM_PITCHING_HR + TEAM_FIELDING_E + TEAM_FIELDING_DP
##
## Df Sum of Sq RSS AIC
## - TEAM_BASERUN_CS 1 0.0129 51.260 -5948.5
## - TEAM_BATTING_HR 1 0.0317 51.279 -5947.9
## <none> 51.247 -5946.9
## - TEAM_BATTING_BB 1 0.0824 51.330 -5946.2
## - TEAM_BATTING_2B 1 0.1959 51.443 -5942.4
## - TEAM_PITCHING_HR 1 0.2204 51.468 -5941.6
## - TEAM_BATTING_3B 1 0.3337 51.581 -5937.9
## - TEAM_PITCHING_H 1 0.3527 51.600 -5937.2
## - TEAM_BASERUN_SB 1 1.1358 52.383 -5911.6
## - TEAM_FIELDING_DP 1 1.4838 52.731 -5900.3
## - TEAM_FIELDING_E 1 3.3164 54.564 -5842.1
## - TEAM_BATTING_H 1 6.1379 57.385 -5756.2
##
## Step: AIC=-5948.5
## log(TARGET_WINS + 1) ~ TEAM_BATTING_H + TEAM_BATTING_2B + TEAM_BATTING_3B +
## TEAM_BATTING_HR + TEAM_BATTING_BB + TEAM_BASERUN_SB + TEAM_PITCHING_H +
## TEAM_PITCHING_HR + TEAM_FIELDING_E + TEAM_FIELDING_DP
##
## Df Sum of Sq RSS AIC
## - TEAM_BATTING_HR 1 0.0288 51.289 -5949.5
## <none> 51.260 -5948.5
## - TEAM_BATTING_BB 1 0.0907 51.351 -5947.5
## - TEAM_BATTING_2B 1 0.2005 51.461 -5943.9
## - TEAM_PITCHING_HR 1 0.2205 51.481 -5943.2
## - TEAM_BATTING_3B 1 0.3362 51.597 -5939.4
## - TEAM_PITCHING_H 1 0.3639 51.624 -5938.4
## - TEAM_BASERUN_SB 1 1.1452 52.406 -5912.9
## - TEAM_FIELDING_DP 1 1.4971 52.757 -5901.4
## - TEAM_FIELDING_E 1 3.4025 54.663 -5841.0
## - TEAM_BATTING_H 1 6.1321 57.392 -5758.0
##
## Step: AIC=-5949.55
## log(TARGET_WINS + 1) ~ TEAM_BATTING_H + TEAM_BATTING_2B + TEAM_BATTING_3B +
## TEAM_BATTING_BB + TEAM_BASERUN_SB + TEAM_PITCHING_H + TEAM_PITCHING_HR +
## TEAM_FIELDING_E + TEAM_FIELDING_DP
##
## Df Sum of Sq RSS AIC
## <none> 51.289 -5949.5
## - TEAM_BATTING_BB 1 0.0828 51.372 -5948.8
## - TEAM_BATTING_2B 1 0.2022 51.491 -5944.8
## - TEAM_PITCHING_H 1 0.3366 51.626 -5940.4
## - TEAM_BATTING_3B 1 0.4030 51.692 -5938.2
## - TEAM_PITCHING_HR 1 0.9628 52.252 -5919.9
## - TEAM_BASERUN_SB 1 1.1686 52.458 -5913.2
## - TEAM_FIELDING_DP 1 1.5021 52.791 -5902.4
## - TEAM_FIELDING_E 1 3.3737 54.663 -5843.0
## - TEAM_BATTING_H 1 6.1100 57.399 -5759.8
plot(stp_logy_model, which=1)
Our Residuals vs. Fitted plot suggests a somewhat linear model but there is visible bowing in the trend line.
plot(stp_logy_model, which = 2)
shapiro.test(residuals(stp_logy_model))
##
## Shapiro-Wilk normality test
##
## data: residuals(stp_logy_model)
## W = 0.98025, p-value = 1.416e-14
# Shapiro-Wilk normality test: look for high p-value
Our QQ plot suggests normality thought there is obvious skewing on the tails, particularly on the left.
A Shapiro Wilk’s Test statistic had a value of 0.980, also suggesting normality. However, since the p-value (1.416e-14) is less than < 0.05 we may still be violating our normality assumption.
plot(stp_logy_model, which = 3)
plot(stp_logy_model, which = 4)
bptest(stp_logy_model)
##
## studentized Breusch-Pagan test
##
## data: stp_logy_model
## BP = 383.07, df = 14, p-value < 2.2e-16
# Breusch-Pagan test; look for high p-value
Our Scale-Location plot shows clear bowing in the trend line, suggesting that the variance of the residuals is not constant and therefore heteroschedastic. While there is no obvious fan/wedge pattern, there is clustering in the center suggesting underfitting, high leverage outliers or that additional transformation may be needed. As our Cook’s Distance plot has no values with a greater than 1, we can rule our the effects of high leverage points. However, it should be pointed the leverage points appear to be more influential than in our backwards selected model or our Box-Cox transformed model.
The Breusch-Page test statistic BP (383.07) indicates a substantial relationship between the residual variance and the predictors and the small p-value (2.2e-16) suggest evidence of heteroscedasticity. Though the values are different that we may need to transform the data to meet the Assumption of Homoscedasticity.
acf(residuals(stp_logy_model))
durbinWatsonTest(stp_logy_model)
## lag Autocorrelation D-W Statistic p-value
## 1 -0.004100597 2.002974 0.946
## Alternative hypothesis: rho != 0
# Durbin Watson should be close to 2
Our Autocorrelation Function shows that there are lags above the blue dashed line, suggesting no autocorrelation. This is confirmed through a Durbin-Watson test statistic value of 2.00 and an autocorrelation value of -0.0041. Furthermore, as our p-value (0.96) is greater than 0.05, we do not have enough evidence to reject the null hypothesis that there is no autocorrelation. In other words, the test results suggest that our model’s residuals are independent and therefore do not violate the Independence Assumption.
Conclusion Applying a log transformation the dependent variable in our model appears to inferior to the backward selected model and the Box-Cox transformed model. We should therefore avoid using this model.
Some of our predictors (TEAM_PITCHING_SO, TEAM_PITCHING_BB, & TEAM_PITCHING_H) showed the presence of outliers. We will transform these variables with a log transformation.
stplg75_train_df <- stp75_train_df %>%
mutate(
TEAM_PITCHING_SO = log(TEAM_PITCHING_SO),
TEAM_PITCHING_BB = log(TEAM_PITCHING_BB),
TEAM_PITCHING_H = log(TEAM_PITCHING_H),
)
stplg25_test_df <- stp25_test_df %>%
mutate(
TEAM_PITCHING_SO = log1p(TEAM_PITCHING_SO),
TEAM_PITCHING_BB = log1p(TEAM_PITCHING_BB),
TEAM_PITCHING_H = log1p(TEAM_PITCHING_H),
)
stplg75_train_df %>%
gather(variable, value, -TARGET_WINS) %>%
ggplot(., aes(value, TARGET_WINS)) +
geom_point(fill = "#628B3A", color="#628B3A") +
geom_smooth(method = "lm", se = FALSE, color = "black") +
facet_wrap(~variable, scales ="free", ncol = 4) +
labs(x = element_blank(), y = "Wins")
## `geom_smooth()` using formula = 'y ~ x'
Our scatter plots show some improvement in our outliers.
We will use backward step-wise selection to build our model.
stp_logx_model_full <- lm(TARGET_WINS ~ ., data = stplg75_train_df)
# Backward step-wise regression
stp_logx_model <- step(stp_logx_model_full, direction = "backward")
## Start: AIC=8695.75
## TARGET_WINS ~ TEAM_BATTING_H + TEAM_BATTING_2B + TEAM_BATTING_3B +
## TEAM_BATTING_HR + TEAM_BATTING_BB + TEAM_BATTING_SO + TEAM_BASERUN_SB +
## TEAM_BASERUN_CS + TEAM_PITCHING_H + TEAM_PITCHING_HR + TEAM_PITCHING_BB +
## TEAM_PITCHING_SO + TEAM_FIELDING_E + TEAM_FIELDING_DP
##
## Df Sum of Sq RSS AIC
## - TEAM_PITCHING_HR 1 7.2 275487 8693.8
## - TEAM_BASERUN_CS 1 148.0 275628 8694.7
## - TEAM_BATTING_HR 1 238.1 275718 8695.2
## <none> 275480 8695.8
## - TEAM_BATTING_2B 1 675.3 276155 8697.9
## - TEAM_PITCHING_H 1 2525.5 278005 8709.3
## - TEAM_BATTING_SO 1 3066.1 278546 8712.6
## - TEAM_PITCHING_SO 1 3646.6 279127 8716.2
## - TEAM_BATTING_3B 1 4252.8 279733 8719.9
## - TEAM_FIELDING_DP 1 6895.7 282376 8735.9
## - TEAM_BATTING_H 1 6934.6 282415 8736.1
## - TEAM_BASERUN_SB 1 9010.2 284490 8748.6
## - TEAM_PITCHING_BB 1 9160.2 284640 8749.5
## - TEAM_BATTING_BB 1 10137.4 285617 8755.3
## - TEAM_FIELDING_E 1 15045.3 290525 8784.4
##
## Step: AIC=8693.8
## TARGET_WINS ~ TEAM_BATTING_H + TEAM_BATTING_2B + TEAM_BATTING_3B +
## TEAM_BATTING_HR + TEAM_BATTING_BB + TEAM_BATTING_SO + TEAM_BASERUN_SB +
## TEAM_BASERUN_CS + TEAM_PITCHING_H + TEAM_PITCHING_BB + TEAM_PITCHING_SO +
## TEAM_FIELDING_E + TEAM_FIELDING_DP
##
## Df Sum of Sq RSS AIC
## - TEAM_BASERUN_CS 1 147.3 275634 8692.7
## <none> 275487 8693.8
## - TEAM_BATTING_2B 1 678.9 276166 8696.0
## - TEAM_PITCHING_H 1 2798.8 278286 8709.0
## - TEAM_BATTING_SO 1 3154.4 278642 8711.2
## - TEAM_BATTING_HR 1 3200.3 278687 8711.5
## - TEAM_PITCHING_SO 1 3847.4 279335 8715.4
## - TEAM_BATTING_3B 1 4347.2 279834 8718.5
## - TEAM_FIELDING_DP 1 6894.3 282381 8733.9
## - TEAM_BATTING_H 1 6928.5 282416 8734.1
## - TEAM_BASERUN_SB 1 9068.5 284556 8747.0
## - TEAM_PITCHING_BB 1 9277.1 284764 8748.2
## - TEAM_BATTING_BB 1 10187.3 285674 8753.7
## - TEAM_FIELDING_E 1 16384.1 291871 8790.2
##
## Step: AIC=8692.71
## TARGET_WINS ~ TEAM_BATTING_H + TEAM_BATTING_2B + TEAM_BATTING_3B +
## TEAM_BATTING_HR + TEAM_BATTING_BB + TEAM_BATTING_SO + TEAM_BASERUN_SB +
## TEAM_PITCHING_H + TEAM_PITCHING_BB + TEAM_PITCHING_SO + TEAM_FIELDING_E +
## TEAM_FIELDING_DP
##
## Df Sum of Sq RSS AIC
## <none> 275634 8692.7
## - TEAM_BATTING_2B 1 713.4 276348 8695.1
## - TEAM_PITCHING_H 1 2705.7 278340 8707.4
## - TEAM_BATTING_SO 1 3248.7 278883 8710.7
## - TEAM_BATTING_HR 1 3608.4 279243 8712.9
## - TEAM_PITCHING_SO 1 3986.6 279621 8715.2
## - TEAM_BATTING_3B 1 4358.6 279993 8717.4
## - TEAM_FIELDING_DP 1 6979.1 282614 8733.3
## - TEAM_BATTING_H 1 7058.8 282693 8733.8
## - TEAM_BASERUN_SB 1 9037.2 284672 8745.7
## - TEAM_PITCHING_BB 1 9222.8 284857 8746.8
## - TEAM_BATTING_BB 1 10245.1 285880 8752.9
## - TEAM_FIELDING_E 1 16416.1 292051 8789.3
Looking at the summary for this model shows that all predictors have high statistic significance.
summary(stp_logx_model)
##
## Call:
## lm(formula = TARGET_WINS ~ TEAM_BATTING_H + TEAM_BATTING_2B +
## TEAM_BATTING_3B + TEAM_BATTING_HR + TEAM_BATTING_BB + TEAM_BATTING_SO +
## TEAM_BASERUN_SB + TEAM_PITCHING_H + TEAM_PITCHING_BB + TEAM_PITCHING_SO +
## TEAM_FIELDING_E + TEAM_FIELDING_DP, data = stplg75_train_df)
##
## Residuals:
## Min 1Q Median 3Q Max
## -52.956 -8.307 -0.153 7.949 61.107
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -41.094143 18.185201 -2.260 0.0240 *
## TEAM_BATTING_H 0.038425 0.005839 6.581 6.23e-11 ***
## TEAM_BATTING_2B -0.021815 0.010427 -2.092 0.0366 *
## TEAM_BATTING_3B 0.103040 0.019926 5.171 2.60e-07 ***
## TEAM_BATTING_HR 0.050302 0.010691 4.705 2.74e-06 ***
## TEAM_BATTING_BB 0.071239 0.008986 7.928 4.01e-15 ***
## TEAM_BATTING_SO -0.023366 0.005234 -4.464 8.56e-06 ***
## TEAM_BASERUN_SB 0.037557 0.005044 7.446 1.53e-13 ***
## TEAM_PITCHING_H 17.554470 4.308701 4.074 4.83e-05 ***
## TEAM_PITCHING_BB -28.690911 3.814242 -7.522 8.71e-14 ***
## TEAM_PITCHING_SO 16.554274 3.347389 4.945 8.35e-07 ***
## TEAM_FIELDING_E -0.037062 0.003693 -10.036 < 2e-16 ***
## TEAM_FIELDING_DP -0.095601 0.014610 -6.543 7.95e-11 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 12.77 on 1691 degrees of freedom
## Multiple R-squared: 0.331, Adjusted R-squared: 0.3263
## F-statistic: 69.72 on 12 and 1691 DF, p-value: < 2.2e-16
The Residuals vs. Fitted and QQ Plots show a fairly linear pattern, while Scale-Location plot suggest Homoscedasticity. The Residuals vs Leverage plot reveals that our outliers have less leverage than in the pre- log transformed model.
plot(stp_logx_model)
stplg75_train_df <- stplg75_train_df |>
mutate(
n = row_number()
)
A VIF Test shows several variables that are highly correlated, including TEAM_PITCHING_H, TEAM_BATTING_BB, TEAM_PITCHING_SO, TEAM_PITCHING_BB, TEAM_FIELDING_E & TEAM_BATTING_H. We may choose to leave these out.
vif(stp_logx_model)
## TEAM_BATTING_H TEAM_BATTING_2B TEAM_BATTING_3B TEAM_BATTING_HR
## 6.803881 2.484158 3.052159 4.407697
## TEAM_BATTING_BB TEAM_BATTING_SO TEAM_BASERUN_SB TEAM_PITCHING_H
## 12.534258 15.179164 1.884433 15.254714
## TEAM_PITCHING_BB TEAM_PITCHING_SO TEAM_FIELDING_E TEAM_FIELDING_DP
## 8.746531 11.608842 6.971147 1.355683
plot(stp_logx_model, which=1)
Our Residuals vs. Fitted plot suggest a fairly linear model.
plot(stp_logx_model, which = 2)
# Shapiro-Wilk normality test: look for high p-value
shapiro.test(residuals(stp_logx_model))
##
## Shapiro-Wilk normality test
##
## data: residuals(stp_logx_model)
## W = 0.99433, p-value = 4.301e-06
Our QQ plot suggests normality thought there is some skewing on the tails, particularly on the right. The skewing appears to be less pitched than in our backward-selected model or Box-Cox Transformed model.
A Shapiro Wilk’s Test statistic had a value of 0.9943, also suggesting normality. However, since the p-value (4.273e-06) is less than < 0.05 we may still be violating our normality assumption.
plot(stp_logx_model, which = 3)
plot(stp_logx_model, which = 4)
bptest(stp_logx_model)
##
## studentized Breusch-Pagan test
##
## data: stp_logx_model
## BP = 302.08, df = 12, p-value < 2.2e-16
# Breusch-Pagan test; look for high p-value
Our Scale-Location plot shows that points appear somewhat evenly distributed above and below the trend line but there is clear bowing in the trend line, suggesting that the variance of the residuals is not constant and therefore heteroschedastic. While there is no obvious fan/wedge pattern, there is clustering in the center suggesting underfitting, high leverage outliers or that additional transformation may be needed. As our Cook’s Distance plot has no values with a greater than 1, we can rule our the effects of high leverage points.
The Breusch-Page test statistic BP (302.21) and the small p-value (2.2e-16) suggest evidence of heteroscedasticity. Though the values are different that we may need to transform the data to meet the Assumption of Homoscedasticity.
acf(residuals(stp_logx_model))
durbinWatsonTest(stp_logx_model)
## lag Autocorrelation D-W Statistic p-value
## 1 -0.0084413 2.011121 0.822
## Alternative hypothesis: rho != 0
# Durbin Watson should be close to 2
Our Autocorrelation Function shows that there are lags above the blue dashed line, suggesting no autocorrelation. This is confirmed through a Durbin-Watson test statistic value of 2.011 and an autocorrelation value of -0.008. Furthermore, as our p-value (0.822) is greater than 0.05, we do not have enough evidence to reject the null hypothesis that there is no autocorrelation. In other words, the test results suggest that our model’s residuals are independent and therefore do not violate the Independence Assumption.
The Least Absolute Shrinkage and Selection Operator (LASSO) selection method uses a shrinkage approach to determining the optimal predictors by attempting to find a balance between simplicity and accuracy.It applies a penalty to the standard linear regression model to encourage the coefficients of features with weak influence to equal zero to prevent overfitting.
x = model.matrix(TARGET_WINS ~ ., data = stplg75_train_df)
y = stplg75_train_df$TARGET_WINS
fit_lasso = glmnet(x, y, alpha=1)
plot(fit_lasso, xvar="lambda", label=TRUE)
cv_lasso = cv.glmnet(x,y,alpha=1)
#plot(cv.lasso)
coef(cv_lasso)
## 17 x 1 sparse Matrix of class "dgCMatrix"
## s1
## (Intercept) 20.682694712
## (Intercept) .
## TEAM_BATTING_H 0.042771197
## TEAM_BATTING_2B .
## TEAM_BATTING_3B 0.025371136
## TEAM_BATTING_HR .
## TEAM_BATTING_BB 0.009857031
## TEAM_BATTING_SO .
## TEAM_BASERUN_SB 0.020057853
## TEAM_BASERUN_CS .
## TEAM_PITCHING_H .
## TEAM_PITCHING_HR 0.016665501
## TEAM_PITCHING_BB .
## TEAM_PITCHING_SO .
## TEAM_FIELDING_E -0.017855099
## TEAM_FIELDING_DP -0.062078149
## n .
Performing LASSO on our training subset – 75% of full training set with log transformation applied – gives us 7 predictors: - TEAM_BATTING_H - TEAM_BATTING_3B - TEAM_BATTING_BB - TEAM_BASERUN_SB - TEAM_PITCHING_HR - TEAM_FIELDING_E - TEAM_FIELDING_DP
lasso_model <- lm(TARGET_WINS ~ TEAM_BATTING_H + TEAM_BATTING_3B + TEAM_BATTING_BB + TEAM_BASERUN_SB + TEAM_PITCHING_HR + TEAM_FIELDING_E + TEAM_FIELDING_DP, data = stplg75_train_df)
summary(lasso_model)
##
## Call:
## lm(formula = TARGET_WINS ~ TEAM_BATTING_H + TEAM_BATTING_3B +
## TEAM_BATTING_BB + TEAM_BASERUN_SB + TEAM_PITCHING_HR + TEAM_FIELDING_E +
## TEAM_FIELDING_DP, data = stplg75_train_df)
##
## Residuals:
## Min 1Q Median 3Q Max
## -54.218 -8.637 0.087 8.262 56.386
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 18.793612 4.011279 4.685 3.02e-06 ***
## TEAM_BATTING_H 0.044628 0.002843 15.700 < 2e-16 ***
## TEAM_BATTING_3B 0.082614 0.018680 4.423 1.04e-05 ***
## TEAM_BATTING_BB 0.009393 0.003782 2.484 0.0131 *
## TEAM_BASERUN_SB 0.027885 0.004659 5.985 2.63e-09 ***
## TEAM_PITCHING_HR 0.044835 0.007878 5.691 1.48e-08 ***
## TEAM_FIELDING_E -0.023203 0.002170 -10.691 < 2e-16 ***
## TEAM_FIELDING_DP -0.106021 0.014819 -7.155 1.24e-12 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 13.03 on 1696 degrees of freedom
## Multiple R-squared: 0.301, Adjusted R-squared: 0.2981
## F-statistic: 104.3 on 7 and 1696 DF, p-value: < 2.2e-16
vif(lasso_model)
## TEAM_BATTING_H TEAM_BATTING_3B TEAM_BATTING_BB TEAM_BASERUN_SB
## 1.547697 2.574600 2.130845 1.543286
## TEAM_PITCHING_HR TEAM_FIELDING_E TEAM_FIELDING_DP
## 2.354824 2.310808 1.338637
Our VIF test shows no strong correlation between predictors.
plot(lasso_model, which=1)
Our Residuals vs. Fitted plot suggest a fairly linear model.
plot(lasso_model, which = 2)
# Shapiro-Wilk normality test: look for high p-value
shapiro.test(residuals(lasso_model))
##
## Shapiro-Wilk normality test
##
## data: residuals(lasso_model)
## W = 0.99644, p-value = 0.0005262
Our QQ plot suggests normality thought there is some skewing on the tails, particularly on the right. The skewing appears to be less pitched than in our backward-selected model or Box-Cox Transformed model.
A Shapiro Wilk’s Test statistic had a value of 0.9964, also suggesting normality. However, since the p-value (0.0005) is less than < 0.05 we may still be violating our normality assumption.
plot(lasso_model, which = 3)
plot(lasso_model, which = 4)
bptest(lasso_model)
##
## studentized Breusch-Pagan test
##
## data: lasso_model
## BP = 209.89, df = 7, p-value < 2.2e-16
# Breusch-Pagan test; look for high p-value
Our Scale-Location plot shows that points appear somewhat evenly distributed above and below the trend line but there is slight bowing in the trend line, suggesting that the variance of the residuals is not constant and therefore heteroschedastic. While there is no obvious fan/wedge pattern, there is clustering in the center suggesting underfitting, high leverage outliers or that additional transformation may be needed. As our Cook’s Distance plot has no values with a greater than 1, we can rule our the effects of high leverage points.
The Breusch-Page test statistic BP (209.89) and the small p-value (2.2e-16) suggest evidence of heteroscedasticity. Though the values are different that we may need to transform the data to meet the Assumption of Homoscedasticity.
acf(residuals(lasso_model))
durbinWatsonTest(lasso_model)
## lag Autocorrelation D-W Statistic p-value
## 1 -0.01774321 2.029976 0.508
## Alternative hypothesis: rho != 0
# Durbin Watson should be close to 2
Our Autocorrelation Function shows that there are lags above the blue dashed line, suggesting no autocorrelation. This is confirmed through a Durbin-Watson test statistic value of 2.03 and an autocorrelation value of -0.0177. Furthermore, as our p-value (0.504) is greater than 0.05, we do not have enough evidence to reject the null hypothesis that there is no autocorrelation. In other words, the test results suggest that our model’s residuals are independent and therefore do not violate the Independence Assumption.
Here we are comparing the all models using our training subset (75% of observations).
#stp_model_inv - is throwing errors due to difference in scales
compare_performance(base_model, high_impact_model, log_model, back_select_model, stp_model_sm, stp_logy_model, stp_logx_model, lasso_model, rank = TRUE)
The “compare_performance” function compares various test results including Adjusted R-Squared, AIC, and BIC. Based on these results, “compare_performance” suggests that our Log Transformed Dependent variable (stp_logy_model_) performed the best. However, this model had the most obvious violations of regression assumptions. Thus we will discard it moving forward. Re-running “compare_performance” without this model shows that
#stp_model_inv - is throwing errors due to difference in scales
compare_performance(base_model, high_impact_model, log_model, back_select_model, stp_model_sm, stp_logx_model, lasso_model)
If interpretability is most important → Use base Stats Model.
If statistical optimization is preferred → Use High-Impact Model.
If non-linearity is a concern → Use Log-Transformed Model.
If statistical significance is most important → Log-Transformed Step-wise Model
In this section, we will use the test dataframe to test the accuracy of our model’s predictions
# Do we need to clean eval dataset first?
# Stats Model predictions
predictedWins = predict(base_model, stp25_test_df)
stp25_test_df["PREDICTED_WINS"] = predictedWins
# Improved Model predictions
predictedWins = predict(improved_model, stp25_test_df)
stp25_test_df["PREDICTED_WINS_IMP"] = predictedWins
# High-Impact Model
predictedWins = predict(high_impact_model, stp25_test_df)
stp25_test_df["PREDICTED_WINS_HIM"] = predictedWins
# Log-Transformed Model
predictedWins = predict(log_model, train_df_log)
train_df_log["PREDICTED_WINS_LOG"] = predictedWins
# Log-Transformed Step-wise Model
predictedWins = predict(stp_logx_model, stplg25_test_df)
stplg25_test_df["PREDICTED_WINS_STP"] = predictedWins
# BoxCox-Transformed Step-wise Model
### TODO: Backward Tranform
#predictedWins = predict(stp_mode_inv, train_df_log)
#train_df_log["PREDICTED_WINS_INV"] = predictedWins
# Create a df to store our results
evaluation_metrics <- data.frame(
model_name = character(0),
MAE = numeric(0),
RMSE = numeric(0),
RSquared = numeric(0)
)
eval_predictions <- function(model, predict_col, target_col, model_name) {
pred = lm(predict_col ~ target_col)
plot(predict_col, target_col, xlab="Actual Wins", ylab="Predicted Wins")
abline(pred)
plot(model$residuals)
model_metrics = data.frame(
model_name = model_name,
MAE = mae(target_col, predict_col),
RMSE = rmse(target_col, predict_col),
RSquared = (cor(target_col, predict_col)^2 )
)
evaluation_metrics <<- rbind(evaluation_metrics, model_metrics)
}
# Evaluate Stats Model predictions
eval_predictions(base_model, stp25_test_df$PREDICTED_WINS, stp25_test_df$TARGET_WINS, "Stats")
# Improved Model
eval_predictions(improved_model, stp25_test_df$PREDICTED_WINS_IMP, stp25_test_df$TARGET_WINS, "Improved")
# Evaluate High-Impact Model
eval_predictions(high_impact_model, stp25_test_df$PREDICTED_WINS_HIM, stp25_test_df$TARGET_WINS, "High-Impact")
# Evaluate Log-Transformed Model
eval_predictions(log_model, train_df_log$PREDICTED_WINS_LOG, train_df_log$TARGET_WINS, "Log-Tranformed")
# Evaluate Log-Transformed Step-wise Model
eval_predictions(stp_logx_model, stplg25_test_df$PREDICTED_WINS_STP, stplg25_test_df$TARGET_WINS, "LT Step-wise")
# TODO: backward transform
# Evaluate Box-Cox Transformed Step-wise Model
#eval_predictions(stp_model_inv, train_df_log$PREDICTED_WINS_INV, train_df_log$TARGET_WINS, "Box-Cox")
To evaluate the accuracy of each our model’s predictions, we will compare: - the Mean Absolute Error (MAE) - the Root Mean Squared Error (RMSE) - the R-squared (R²)
print(evaluation_metrics)
## model_name MAE RMSE RSquared
## 1 Stats 10.96006 14.00786 0.2234439
## 2 Improved 10.69973 13.76642 0.2507960
## 3 High-Impact 10.89616 13.95167 0.2292489
## 4 Log-Tranformed 10.81108 13.57413 0.2379521
## 5 LT Step-wise 10.12808 13.15547 0.3175279
A quick examination of these shows the Log-Transformed Step-wise Model has the lowest MAE and RMSE as well as the highest R-squared, suggesting that this model is somewhat more accurate than the other four evaluated. For confirmation we will perform a 5-fold cross-validation below:
library(caret)
train_control <- trainControl(method = "cv", number = 5) # 10-fold cross-validation
model_cv <- train(TARGET_WINS ~ TEAM_BATTING_H + TEAM_BATTING_2B +
TEAM_BATTING_3B + TEAM_BATTING_HR + TEAM_BATTING_BB + TEAM_BATTING_SO +
TEAM_BASERUN_SB + TEAM_PITCHING_H + TEAM_PITCHING_BB + TEAM_PITCHING_SO +
TEAM_FIELDING_E + TEAM_FIELDING_DP, data = stplg75_train_df, method = "lm", trControl = train_control)
print(model_cv)
## Linear Regression
##
## 1704 samples
## 12 predictor
##
## No pre-processing
## Resampling: Cross-Validated (5 fold)
## Summary of sample sizes: 1363, 1364, 1363, 1363, 1363
## Resampling results:
##
## RMSE Rsquared MAE
## 12.90372 0.3150703 10.02132
##
## Tuning parameter 'intercept' was held constant at a value of TRUE
summary(stp_logx_model)
##
## Call:
## lm(formula = TARGET_WINS ~ TEAM_BATTING_H + TEAM_BATTING_2B +
## TEAM_BATTING_3B + TEAM_BATTING_HR + TEAM_BATTING_BB + TEAM_BATTING_SO +
## TEAM_BASERUN_SB + TEAM_PITCHING_H + TEAM_PITCHING_BB + TEAM_PITCHING_SO +
## TEAM_FIELDING_E + TEAM_FIELDING_DP, data = stplg75_train_df)
##
## Residuals:
## Min 1Q Median 3Q Max
## -52.956 -8.307 -0.153 7.949 61.107
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -41.094143 18.185201 -2.260 0.0240 *
## TEAM_BATTING_H 0.038425 0.005839 6.581 6.23e-11 ***
## TEAM_BATTING_2B -0.021815 0.010427 -2.092 0.0366 *
## TEAM_BATTING_3B 0.103040 0.019926 5.171 2.60e-07 ***
## TEAM_BATTING_HR 0.050302 0.010691 4.705 2.74e-06 ***
## TEAM_BATTING_BB 0.071239 0.008986 7.928 4.01e-15 ***
## TEAM_BATTING_SO -0.023366 0.005234 -4.464 8.56e-06 ***
## TEAM_BASERUN_SB 0.037557 0.005044 7.446 1.53e-13 ***
## TEAM_PITCHING_H 17.554470 4.308701 4.074 4.83e-05 ***
## TEAM_PITCHING_BB -28.690911 3.814242 -7.522 8.71e-14 ***
## TEAM_PITCHING_SO 16.554274 3.347389 4.945 8.35e-07 ***
## TEAM_FIELDING_E -0.037062 0.003693 -10.036 < 2e-16 ***
## TEAM_FIELDING_DP -0.095601 0.014610 -6.543 7.95e-11 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 12.77 on 1691 degrees of freedom
## Multiple R-squared: 0.331, Adjusted R-squared: 0.3263
## F-statistic: 69.72 on 12 and 1691 DF, p-value: < 2.2e-16
MAE and RMSE is even lower when using cross-validation. Although our R-squared value decreased from 0.3183 to 0.3034024, this value is still higher than the model with second highest R-squared value of 0.2368 for the Log-Tranformed model.
Our previous test suggest that the Log-Transformed Step-wise Model appears to be the most accurate at predicting wins over a team’s season. However, with 12 predictors, this model is fairly complex. Given the audience for our report and the need to clearly describe our model to the coach, we should select a simpler model that is still statistically significant and fairly accurate. Therefore, we will select our Improved model.
The Improved Model is the best choice for predicting TARGET_WINS because it has the highest R² (27.89%), the lowest residual standard error (13.25), and avoids severe multicollinearity issues.
Unlike the High-Impact Model, which suffers from high VIF values (HR & Pitching HR > 20), and the Log Model, which has weaker predictive power (R² = 23.41%), the Improved Model balances performance, interpretability, and statistical significance.
Key predictors like hits, home runs, strikeouts, and fielding errors are logical, and the interaction between home runs and walks (HR * BB) is highly significant, confirming that plate discipline enhances home run effectiveness.
The only concern is TEAM_BATTING_2B (Doubles), which has an unexpected negative coefficient and needs further analysis.
Additionally, the Improved Model satisfies all key assumptions of multiple linear regression—it demonstrates linearity, independence of errors, normality of residuals, and no severe multicollinearity (all VIF values < 5).
While a simpler model is preferred, the slight increase in complexity is justified by better accuracy and logical relationships.
To finalize the model, we should re-run it without TEAM_BATTING_2B to see if it improves further, perform residual diagnostics, and validate with cross-validation. Given its strong balance of accuracy and interpretability, the Improved Model is the best option for predicting team wins.
Making Predictions Using the Evaluation Dataset:
missing_values <- eval_df %>%
summarise(across(everything(), ~ sum(is.na(.)))) %>%
pivot_longer(cols = everything(), names_to = "Variable", values_to = "Missing_Count")
print(missing_values)
## # A tibble: 15 × 2
## Variable Missing_Count
## <chr> <int>
## 1 TEAM_BATTING_H 0
## 2 TEAM_BATTING_2B 0
## 3 TEAM_BATTING_3B 0
## 4 TEAM_BATTING_HR 0
## 5 TEAM_BATTING_BB 0
## 6 TEAM_BATTING_SO 18
## 7 TEAM_BASERUN_SB 13
## 8 TEAM_BASERUN_CS 87
## 9 TEAM_BATTING_HBP 240
## 10 TEAM_PITCHING_H 0
## 11 TEAM_PITCHING_HR 0
## 12 TEAM_PITCHING_BB 0
## 13 TEAM_PITCHING_SO 18
## 14 TEAM_FIELDING_E 0
## 15 TEAM_FIELDING_DP 31
eval_df <- eval_df[, !names(eval_df) %in% "INDEX"]
eval_df <- eval_df[, !names(eval_df) %in% "TEAM_BATTING_HBP"]
eval_df
Instead of removing missing values, fill them with the mean or median:
eval_df$TEAM_BATTING_SO[is.na(eval_df$TEAM_BATTING_SO)] <- mean(eval_df$TEAM_BATTING_SO, na.rm = TRUE)
eval_df$TEAM_BASERUN_SB[is.na(eval_df$TEAM_BASERUN_SB)] <- mean(eval_df$TEAM_BASERUN_SB, na.rm = TRUE)
eval_df$TEAM_BASERUN_CS[is.na(eval_df$TEAM_BASERUN_CS)] <- mean(eval_df$TEAM_BASERUN_CS, na.rm = TRUE)
eval_df$TEAM_FIELDING_DP[is.na(eval_df$TEAM_FIELDING_DP)] <- mean(eval_df$TEAM_FIELDING_DP, na.rm = TRUE)
eval_df$TEAM_PITCHING_SO[is.na(eval_df$TEAM_PITCHING_SO)] <- mean(eval_df$TEAM_PITCHING_SO, na.rm = TRUE)
sum(is.na(eval_df)) # Total missing values
## [1] 0
colSums(is.na(eval_df)) # Missing values per column
## TEAM_BATTING_H TEAM_BATTING_2B TEAM_BATTING_3B TEAM_BATTING_HR
## 0 0 0 0
## TEAM_BATTING_BB TEAM_BATTING_SO TEAM_BASERUN_SB TEAM_BASERUN_CS
## 0 0 0 0
## TEAM_PITCHING_H TEAM_PITCHING_HR TEAM_PITCHING_BB TEAM_PITCHING_SO
## 0 0 0 0
## TEAM_FIELDING_E TEAM_FIELDING_DP
## 0 0
skim(eval_df)
| Name | eval_df |
| Number of rows | 259 |
| Number of columns | 14 |
| _______________________ | |
| Column type frequency: | |
| numeric | 14 |
| ________________________ | |
| Group variables | None |
Variable type: numeric
| skim_variable | n_missing | complete_rate | mean | sd | p0 | p25 | p50 | p75 | p100 | hist |
|---|---|---|---|---|---|---|---|---|---|---|
| TEAM_BATTING_H | 0 | 1 | 1469.39 | 150.66 | 819 | 1387.0 | 1455.00 | 1548.0 | 2170 | ▁▂▇▁▁ |
| TEAM_BATTING_2B | 0 | 1 | 241.32 | 49.52 | 44 | 210.0 | 239.00 | 278.5 | 376 | ▁▂▇▇▂ |
| TEAM_BATTING_3B | 0 | 1 | 55.91 | 27.14 | 14 | 35.0 | 52.00 | 72.0 | 155 | ▇▇▃▁▁ |
| TEAM_BATTING_HR | 0 | 1 | 95.63 | 56.33 | 0 | 44.5 | 101.00 | 135.5 | 242 | ▆▅▇▃▁ |
| TEAM_BATTING_BB | 0 | 1 | 498.96 | 120.59 | 15 | 436.5 | 509.00 | 565.5 | 792 | ▁▁▅▇▁ |
| TEAM_BATTING_SO | 0 | 1 | 709.34 | 234.48 | 0 | 565.0 | 709.34 | 904.5 | 1268 | ▁▃▇▆▂ |
| TEAM_BASERUN_SB | 0 | 1 | 123.70 | 91.00 | 0 | 60.5 | 96.00 | 149.0 | 580 | ▇▅▁▁▁ |
| TEAM_BASERUN_CS | 0 | 1 | 52.32 | 18.81 | 0 | 44.0 | 52.32 | 56.0 | 154 | ▁▇▂▁▁ |
| TEAM_PITCHING_H | 0 | 1 | 1813.46 | 1662.91 | 1155 | 1426.5 | 1515.00 | 1681.0 | 22768 | ▇▁▁▁▁ |
| TEAM_PITCHING_HR | 0 | 1 | 102.15 | 57.65 | 0 | 52.0 | 104.00 | 142.5 | 336 | ▇▇▆▁▁ |
| TEAM_PITCHING_BB | 0 | 1 | 552.42 | 172.95 | 136 | 471.0 | 526.00 | 606.5 | 2008 | ▆▇▁▁▁ |
| TEAM_PITCHING_SO | 0 | 1 | 799.67 | 611.78 | 0 | 622.5 | 782.00 | 927.5 | 9963 | ▇▁▁▁▁ |
| TEAM_FIELDING_E | 0 | 1 | 249.75 | 230.90 | 73 | 131.0 | 163.00 | 252.0 | 1568 | ▇▁▁▁▁ |
| TEAM_FIELDING_DP | 0 | 1 | 146.06 | 24.28 | 69 | 134.5 | 146.06 | 160.5 | 204 | ▁▂▇▆▁ |
# Generate predictions using the trained Improved Model
eval_df$PREDICTED_WINS <- predict(improved_model, newdata = eval_df)
# View a sample of predictions
head(eval_df$PREDICTED_WINS)
## [1] 69.20074 71.69295 79.61681 81.88350 78.60632 74.74956
The predicted values represent the expected number of wins for teams based on their batting, pitching, and fielding statistics using the Improved Model.
For example:
A team with a predicted 69.15 wins is expected to win around 69 games in a full season. A team with a predicted 82.52 wins is expected to perform better, likely finishing with around 82-83 wins. The variation in predictions suggests that some teams are stronger than others based on key performance metrics (hits, home runs, walks, strikeouts, fielding errors, etc.).
Now that we have predicted TARGET_WINS, we need to compare these predictions with the actual values in the TARGET_WINS column to assess model accuracy.
# Check structure of the dataset
str(eval_df)
## 'data.frame': 259 obs. of 15 variables:
## $ TEAM_BATTING_H : int 1209 1221 1395 1539 1445 1431 1430 1385 1259 1397 ...
## $ TEAM_BATTING_2B : int 170 151 183 309 203 236 219 158 177 212 ...
## $ TEAM_BATTING_3B : int 33 29 29 29 68 53 55 42 78 42 ...
## $ TEAM_BATTING_HR : int 83 88 93 159 5 10 37 33 23 58 ...
## $ TEAM_BATTING_BB : int 447 516 509 486 95 215 568 356 466 452 ...
## $ TEAM_BATTING_SO : num 1080 929 816 914 416 377 527 609 689 584 ...
## $ TEAM_BASERUN_SB : num 62 54 59 148 124 ...
## $ TEAM_BASERUN_CS : num 50 39 47 57 52.3 ...
## $ TEAM_PITCHING_H : int 1209 1221 1395 1539 3902 2793 1544 1626 1342 1489 ...
## $ TEAM_PITCHING_HR: int 83 88 93 159 14 20 40 39 25 62 ...
## $ TEAM_PITCHING_BB: int 447 516 509 486 257 420 613 418 497 482 ...
## $ TEAM_PITCHING_SO: num 1080 929 816 914 1123 ...
## $ TEAM_FIELDING_E : int 140 135 156 124 616 572 490 328 226 184 ...
## $ TEAM_FIELDING_DP: num 156 164 153 154 130 ...
## $ PREDICTED_WINS : num 69.2 71.7 79.6 81.9 78.6 ...
# Check if ACTUAL_WINS and PREDICTED_WINS are numeric
is.numeric(eval_df$ACTUAL_WINS)
## [1] FALSE
is.numeric(eval_df$PREDICTED_WINS)
## [1] TRUE
# Add a row index to both datasets
train_df$INDEX <- seq_len(nrow(train_df))
eval_df$INDEX <- seq_len(nrow(eval_df))
# First, check if both datasets have a common identifier (like INDEX)
head(train_df$INDEX)
## [1] 1 2 3 4 5 6
head(eval_df$INDEX)
## [1] 1 2 3 4 5 6
# Merge train_df and eval_df using INDEX (if applicable)
merged_df <- merge(train_df[, c("INDEX", "TARGET_WINS")],
eval_df[, c("INDEX", "PREDICTED_WINS")],
by = "INDEX", all = FALSE)
# Now check if both columns have equal length
nrow(merged_df)
## [1] 259
Ensure data consistency before calculating accuracy metrics. Convert to numeric to avoid errors in correlation or calculations.
# Compute MAE, RMSE, and R²
mae <- mae(merged_df$TARGET_WINS, merged_df$PREDICTED_WINS)
rmse <- rmse(merged_df$TARGET_WINS, merged_df$PREDICTED_WINS)
r_squared <- cor(merged_df$TARGET_WINS, merged_df$PREDICTED_WINS)^2
# Print results
cat("Model Accuracy Metrics:\n")
## Model Accuracy Metrics:
cat("Mean Absolute Error (MAE):", round(mae, 2), "\n")
## Mean Absolute Error (MAE): 13.73
cat("Root Mean Squared Error (RMSE):", round(rmse, 2), "\n")
## Root Mean Squared Error (RMSE): 17.14
cat("R-Squared (R²):", round(r_squared, 4), "\n")
## R-Squared (R²): 0.0017
The evaluation results indicate that our Improved Model is not performing well in predicting TARGET_WINS.
The Mean Absolute Error (MAE) of 13.87 suggests that, on average, predictions deviate by about 14 wins per team, which is quite high.
Additionally, the Root Mean Squared Error (RMSE) of 17.38 implies that some predictions have even larger errors, highlighting potential inconsistencies or missing key predictors.
Most concerning is the R-Squared (R²) value of 0.001, which means that the model explains almost none of the variation in team wins—essentially, the predictions are no better than random guesses.
This suggests that the model may be overfitting the training data and failing to generalize, or that it lacks critical predictive features.
To improve performance, we should consider adding more relevant predictors (such as ERA or OBP), removing unimportant or noisy variables, and potentially using alternative mo
With our models tested and evaluated, we can no apply our model to our final evaluation dataframe.
predictedWins = predict(stp_logx_model, eval_df)
eval_df["PREDICTED_WINS"] = predictedWins